Session 10: Data Science London Housing Project
Ding Linli
2021-12-25
set.seed(1) # to return the same result in the following chunks1 Introduction and learning objectives
2 Load data
There are two sets of data, i) training data that has the actual prices ii) out of sample data that has the asking prices. Load both data sets.
Make sure you understand what information each column contains. Note that not all information provided might be useful in predicting house prices, but do not make any assumptions before you decide what information you use in your prediction algorithms.
#read in the data
london_house_prices_2019_training<-read.csv("training_data_assignment_with_prices.csv")
london_house_prices_2019_out_of_sample<-read.csv("test_data_assignment.csv")
#fix data types in both data sets
#fix dates
london_house_prices_2019_training <- london_house_prices_2019_training %>% mutate(date=as.Date(date))
london_house_prices_2019_out_of_sample<-london_house_prices_2019_out_of_sample %>% mutate(date=as.Date(date))
#change characters to factors
london_house_prices_2019_training <- london_house_prices_2019_training %>% mutate_if(is.character,as.factor)
london_house_prices_2019_out_of_sample<-london_house_prices_2019_out_of_sample %>% mutate_if(is.character,as.factor)
#take a quick look at what's in the data
str(london_house_prices_2019_training)## 'data.frame': 13998 obs. of 37 variables:
## $ ID : int 2 3 4 5 7 8 9 10 11 12 ...
## $ date : Date, format: "2019-11-01" "2019-08-08" ...
## $ postcode : Factor w/ 12635 levels "BR1 1AB","BR1 1LR",..: 10897 11027 11264 2031 11241 11066 421 9594 9444 873 ...
## $ property_type : Factor w/ 4 levels "D","F","S","T": 2 2 3 2 3 2 1 4 4 2 ...
## $ whether_old_or_new : Factor w/ 2 levels "N","Y": 1 1 1 1 1 1 1 1 1 1 ...
## $ freehold_or_leasehold : Factor w/ 2 levels "F","L": 2 2 1 2 1 2 1 1 1 2 ...
## $ address1 : Factor w/ 2825 levels "1","1 - 2","1 - 3",..: 2503 792 253 789 569 234 264 418 5 274 ...
## $ address2 : Factor w/ 434 levels "1","10","101",..: 372 NA NA NA NA NA NA NA NA NA ...
## $ address3 : Factor w/ 8543 levels "ABBERTON WALK",..: 6990 6821 3715 2492 4168 2879 3620 5251 6045 6892 ...
## $ town : Factor w/ 133 levels "ABBEY WOOD","ACTON",..: NA NA NA 78 NA NA NA NA NA NA ...
## $ local_aut : Factor w/ 69 levels "ASHFORD","BARKING",..: 36 46 24 36 24 46 65 36 36 17 ...
## $ county : Factor w/ 33 levels "BARKING AND DAGENHAM",..: 22 27 18 25 18 27 5 27 32 8 ...
## $ postcode_short : Factor w/ 247 levels "BR1","BR2","BR3",..: 190 194 198 28 198 194 4 169 167 8 ...
## $ current_energy_rating : Factor w/ 6 levels "B","C","D","E",..: 4 3 3 4 3 2 4 3 4 2 ...
## $ total_floor_area : num 30 50 100 39 88 101 136 148 186 65 ...
## $ number_habitable_rooms : int 2 2 5 2 4 4 6 6 6 3 ...
## $ co2_emissions_current : num 2.3 3 3.7 2.8 3.9 3.1 8.1 5.6 10 1.5 ...
## $ co2_emissions_potential : num 1.7 1.7 1.5 1.1 1.4 1.4 4.1 2 6.1 1.5 ...
## $ energy_consumption_current : int 463 313 212 374 251 175 339 216 308 128 ...
## $ energy_consumption_potential: int 344 175 82 144 90 77 168 75 186 128 ...
## $ windows_energy_eff : Factor w/ 5 levels "Average","Good",..: 1 1 1 5 1 1 1 1 5 1 ...
## $ tenure : Factor w/ 3 levels "owner-occupied",..: 1 2 1 2 1 1 1 2 1 1 ...
## $ latitude : num 51.5 51.5 51.5 51.6 51.5 ...
## $ longitude : num -0.1229 -0.2828 -0.4315 0.0423 -0.4293 ...
## $ population : int 34 75 83 211 73 51 25 91 60 97 ...
## $ altitude : int 8 9 25 11 21 11 95 7 7 106 ...
## $ london_zone : int 1 3 5 3 6 6 3 2 2 3 ...
## $ nearest_station : Factor w/ 592 levels "abbey road","abbey wood",..: 478 358 235 319 180 502 566 30 32 566 ...
## $ water_company : Factor w/ 5 levels "Affinity Water",..: 5 5 1 5 1 5 5 5 5 5 ...
## $ average_income : int 57200 61900 50600 45400 49000 56200 57200 65600 50400 52300 ...
## $ district : Factor w/ 33 levels "Barking and Dagenham",..: 22 27 18 26 18 27 5 27 32 8 ...
## $ price : num 360000 408500 499950 259999 395000 ...
## $ type_of_closest_station : Factor w/ 3 levels "light_rail","rail",..: 3 2 3 1 3 2 1 3 1 1 ...
## $ num_tube_lines : int 1 0 1 0 1 0 0 2 0 0 ...
## $ num_rail_lines : int 0 1 1 0 1 1 0 0 1 0 ...
## $ num_light_rail_lines : int 0 0 0 1 0 0 1 0 1 1 ...
## $ distance_to_station : num 0.528 0.77 0.853 0.29 1.073 ...str(london_house_prices_2019_out_of_sample)## 'data.frame': 1999 obs. of 37 variables:
## $ ID : int 14434 12562 8866 10721 1057 1527 13961 12108 9363 1155 ...
## $ date : Date, format: NA NA ...
## $ postcode : logi NA NA NA NA NA NA ...
## $ property_type : Factor w/ 4 levels "D","F","S","T": 1 2 2 3 4 3 2 3 2 4 ...
## $ whether_old_or_new : Factor w/ 2 levels "N","Y": 1 1 1 1 1 1 1 1 1 1 ...
## $ freehold_or_leasehold : Factor w/ 2 levels "F","L": 1 2 2 1 1 1 2 1 2 1 ...
## $ address1 : logi NA NA NA NA NA NA ...
## $ address2 : logi NA NA NA NA NA NA ...
## $ address3 : logi NA NA NA NA NA NA ...
## $ town : Factor w/ 54 levels "ACTON","ADDISCOMBE",..: NA NA NA NA NA NA NA NA NA NA ...
## $ local_aut : logi NA NA NA NA NA NA ...
## $ county : logi NA NA NA NA NA NA ...
## $ postcode_short : Factor w/ 221 levels "BR1","BR2","BR3",..: 82 50 37 52 214 150 159 115 175 126 ...
## $ current_energy_rating : Factor w/ 6 levels "B","C","D","E",..: 3 2 3 3 4 4 4 3 4 3 ...
## $ total_floor_area : num 150 59 58 74 97.3 ...
## $ number_habitable_rooms : int 6 2 2 5 5 5 5 4 2 5 ...
## $ co2_emissions_current : num 7.3 1.5 2.8 3.5 6.5 4.9 5.1 2.9 4.2 4.3 ...
## $ co2_emissions_potential : num 2.4 1.4 1.2 1.2 5.7 1.6 3 0.8 3.2 2.5 ...
## $ energy_consumption_current : int 274 142 253 256 303 309 240 224 458 253 ...
## $ energy_consumption_potential: int 89 136 110 80 266 101 140 58 357 143 ...
## $ windows_energy_eff : Factor w/ 5 levels "Average","Good",..: 1 1 1 1 1 1 3 1 3 1 ...
## $ tenure : Factor w/ 3 levels "owner-occupied",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ latitude : num 51.6 51.6 51.5 51.6 51.5 ...
## $ longitude : num -0.129 -0.2966 -0.0328 -0.3744 -0.2576 ...
## $ population : int 87 79 23 73 100 24 22 49 65 98 ...
## $ altitude : int 63 38 17 39 8 46 26 16 14 18 ...
## $ london_zone : int 4 4 2 5 2 4 3 6 1 3 ...
## $ nearest_station : Factor w/ 494 levels "abbey wood","acton central",..: 16 454 181 302 431 142 20 434 122 212 ...
## $ water_company : Factor w/ 4 levels "Affinity Water",..: 4 1 4 1 4 4 4 2 4 4 ...
## $ average_income : int 61300 48900 46200 52200 60700 59600 64000 48100 56600 53500 ...
## $ district : Factor w/ 32 levels "Barking and Dagenham",..: 9 4 29 14 17 10 31 15 19 22 ...
## $ type_of_closest_station : Factor w/ 3 levels "light_rail","rail",..: 3 3 1 2 3 2 3 3 3 2 ...
## $ num_tube_lines : int 1 2 0 0 2 0 1 1 2 0 ...
## $ num_rail_lines : int 0 1 0 1 0 1 1 0 0 1 ...
## $ num_light_rail_lines : int 0 1 1 0 0 0 0 1 0 0 ...
## $ distance_to_station : num 0.839 0.104 0.914 0.766 0.449 ...
## $ asking_price : num 750000 229000 152000 379000 930000 350000 688000 386000 534000 459000 ...Additional data features will be added as below. The two chosen features are school_number and crime_number. The school data is from 2016. The crime number is the total crime as of year 2020.
# number of school data source https://data.gov.uk/dataset/6b776872-c786-4960-af1d-dab521aa4ab0/london-schools-atlas
education <- read_sf(here("data/All_schools_shp/school_data_london_Atlas_2016.shp"))
borough_education <- education %>%
group_by(LA_NAME) %>%
summarise(school_number = n())
borough_education <- as.data.frame(borough_education) %>%
select(-geometry)
# training add education feature
training_education <- london_house_prices_2019_training %>%
left_join(borough_education, by=c("district" = "LA_NAME"))
# out-of-sample add education feature
testing_education <- london_house_prices_2019_out_of_sample %>%
left_join(borough_education, by=c("district" = "LA_NAME"))#crime data source https://data.london.gov.uk/dataset/recorded_crime_summary
crime <- read_csv(here("data/MPS Borough Level Crime (most recent 24 months).csv"))
colnames(crime)[5:16] = c("crime202001", "crime202002", "crime202003", "crime202004", "crime202005", "crime202006", "crime202007", "crime202008", "crime202009", "crime202010", "crime202011", "crime202012")
number <- crime %>%
select(crime202001, crime202002, crime202003, crime202004, crime202005, crime202006, crime202007, crime202008, crime202009, crime202010, crime202011, crime202012)
number_crime <- number %>%
mutate(crime_number = rowSums(number))
borough_crime <- as.data.frame(cbind(crime, number_crime$crime_number))
colnames(borough_crime)[28] = c("crime_number")
borough_crime <- borough_crime %>%
select(LookUp_BoroughName, crime_number) %>%
group_by(LookUp_BoroughName) %>%
summarise(crime_number = sum(crime_number))
# training add crime feature
london_house_prices_2019_training <- training_education %>%
left_join(borough_crime, by=c("district" = "LookUp_BoroughName")) %>%
filter(!is.na(crime_number)) %>%
filter(!is.na(population))
# training set scale the useful numeric data
features_needs_scaling <- london_house_prices_2019_training %>%
select(total_floor_area, co2_emissions_current, energy_consumption_current, latitude, longitude, london_zone, average_income, num_light_rail_lines, num_rail_lines, num_tube_lines, distance_to_station, school_number, crime_number)
features_needs_scaling <- scale(features_needs_scaling, center=FALSE, scale=TRUE)
temp <- london_house_prices_2019_training %>%
select(-colnames(features_needs_scaling))
london_house_prices_2019_training <- as.data.frame(cbind(temp,features_needs_scaling))
# out-of-sample set add crime feature
london_house_prices_2019_out_of_sample <- testing_education %>%
left_join(borough_crime, by=c("district" = "LookUp_BoroughName")) %>%
filter(!is.na(crime_number)) %>%
filter(!is.na(population))
# out-of-sample set scale the useful numeric data
features_needs_scaling <- london_house_prices_2019_out_of_sample %>%
select(total_floor_area, co2_emissions_current, energy_consumption_current, latitude, longitude, london_zone, average_income, num_light_rail_lines, num_rail_lines, num_tube_lines, distance_to_station, school_number, crime_number)
features_needs_scaling <- scale(features_needs_scaling, center=FALSE, scale=TRUE)
temp <- london_house_prices_2019_out_of_sample %>%
select(-colnames(features_needs_scaling))
london_house_prices_2019_out_of_sample <- as.data.frame(cbind(temp,features_needs_scaling))skim(london_house_prices_2019_training)| Name | london_house_prices_2019_… |
| Number of rows | 13923 |
| Number of columns | 39 |
| _______________________ | |
| Column type frequency: | |
| character | 1 |
| Date | 1 |
| factor | 17 |
| numeric | 20 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| district | 0 | 1 | 5 | 22 | 0 | 32 | 0 |
Variable type: Date
| skim_variable | n_missing | complete_rate | min | max | median | n_unique |
|---|---|---|---|---|---|---|
| date | 0 | 1 | 2019-01-02 | 2019-12-30 | 2019-07-22 | 262 |
Variable type: factor
| skim_variable | n_missing | complete_rate | ordered | n_unique | top_counts |
|---|---|---|---|---|---|
| postcode | 0 | 1.00 | FALSE | 12566 | E11: 5, BR1: 4, CR0: 4, E17: 4 |
| property_type | 0 | 1.00 | FALSE | 4 | F: 5337, T: 4993, S: 2777, D: 816 |
| whether_old_or_new | 0 | 1.00 | FALSE | 2 | N: 13916, Y: 7 |
| freehold_or_leasehold | 0 | 1.00 | FALSE | 2 | F: 8446, L: 5477 |
| address1 | 0 | 1.00 | FALSE | 2787 | 3: 220, 7: 211, 4: 204, 12: 197 |
| address2 | 10797 | 0.22 | FALSE | 425 | FLA: 216, FLA: 213, FLA: 211, FLA: 159 |
| address3 | 0 | 1.00 | FALSE | 8508 | LON: 25, GRE: 24, THE: 23, MAN: 21 |
| town | 13309 | 0.04 | FALSE | 133 | WAL: 35, CHE: 33, STR: 27, CHI: 24 |
| local_aut | 0 | 1.00 | FALSE | 69 | LON: 7462, ROM: 394, BRO: 274, CRO: 241 |
| county | 0 | 1.00 | FALSE | 33 | BRO: 856, CRO: 723, WAN: 696, HAV: 670 |
| postcode_short | 0 | 1.00 | FALSE | 244 | CR0: 241, SW1: 202, E17: 192, SW1: 187 |
| current_energy_rating | 0 | 1.00 | FALSE | 6 | D: 7066, C: 3452, E: 2638, F: 359 |
| windows_energy_eff | 0 | 1.00 | FALSE | 5 | Ave: 7794, Goo: 3199, Ver: 1696, Poo: 1229 |
| tenure | 0 | 1.00 | FALSE | 3 | own: 11195, ren: 2471, ren: 257 |
| nearest_station | 0 | 1.00 | FALSE | 591 | rom: 199, cha: 102, bex: 98, har: 96 |
| water_company | 0 | 1.00 | FALSE | 5 | Tha: 10410, Aff: 1573, Ess: 1149, SES: 787 |
| type_of_closest_station | 0 | 1.00 | FALSE | 3 | rai: 6482, tub: 4681, lig: 2760 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| ID | 0 | 1 | 8002.24 | 4622.93 | 2.00 | 4000.50 | 8017.00 | 12020.50 | 1.5996e+04 | ▇▇▇▇▇ |
| number_habitable_rooms | 0 | 1 | 4.31 | 1.66 | 1.00 | 3.00 | 4.00 | 5.00 | 1.4000e+01 | ▆▇▁▁▁ |
| co2_emissions_potential | 0 | 1 | 2.22 | 1.44 | -0.50 | 1.30 | 1.80 | 2.70 | 2.0500e+01 | ▇▁▁▁▁ |
| energy_consumption_potential | 0 | 1 | 141.26 | 77.68 | -49.00 | 89.00 | 122.00 | 167.00 | 9.2900e+02 | ▇▃▁▁▁ |
| population | 0 | 1 | 83.74 | 43.70 | 1.00 | 52.00 | 79.00 | 109.00 | 5.1000e+02 | ▇▃▁▁▁ |
| altitude | 0 | 1 | 36.59 | 25.99 | 0.00 | 16.00 | 32.00 | 51.00 | 2.3900e+02 | ▇▃▁▁▁ |
| price | 0 | 1 | 594106.58 | 519788.78 | 77000.00 | 351250.00 | 460000.00 | 650000.00 | 1.0800e+07 | ▇▁▁▁▁ |
| total_floor_area | 0 | 1 | 0.90 | 0.44 | 0.20 | 0.62 | 0.81 | 1.05 | 4.6600e+00 | ▇▂▁▁▁ |
| co2_emissions_current | 0 | 1 | 0.87 | 0.49 | 0.02 | 0.55 | 0.78 | 1.07 | 9.0100e+00 | ▇▁▁▁▁ |
| energy_consumption_current | 0 | 1 | 0.94 | 0.33 | 0.04 | 0.72 | 0.89 | 1.10 | 4.6400e+00 | ▇▅▁▁▁ |
| latitude | 0 | 1 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.0000e+00 | ▂▇▇▇▂ |
| longitude | 0 | 1 | -0.55 | 0.83 | -2.56 | -1.08 | -0.56 | 0.01 | 1.5300e+00 | ▂▅▇▅▂ |
| london_zone | 0 | 1 | 0.93 | 0.36 | 0.25 | 0.75 | 0.99 | 1.24 | 1.7400e+00 | ▇▇▇▆▅ |
| average_income | 0 | 1 | 0.99 | 0.15 | 0.64 | 0.88 | 0.98 | 1.08 | 1.5200e+00 | ▃▇▆▂▁ |
| num_light_rail_lines | 0 | 1 | 0.49 | 0.87 | 0.00 | 0.00 | 0.00 | 0.00 | 2.0500e+00 | ▇▁▁▁▂ |
| num_rail_lines | 0 | 1 | 0.75 | 0.66 | 0.00 | 0.00 | 1.30 | 1.30 | 2.6000e+00 | ▆▁▇▁▁ |
| num_tube_lines | 0 | 1 | 0.52 | 0.85 | 0.00 | 0.00 | 0.00 | 1.17 | 7.0500e+00 | ▇▁▁▁▁ |
| distance_to_station | 0 | 1 | 0.85 | 0.53 | 0.00 | 0.49 | 0.74 | 1.09 | 7.3200e+00 | ▇▁▁▁▁ |
| school_number | 0 | 1 | 0.97 | 0.24 | 0.57 | 0.85 | 0.93 | 1.11 | 1.5200e+00 | ▃▇▆▃▂ |
| crime_number | 0 | 1 | 0.96 | 0.29 | 0.45 | 0.71 | 0.99 | 1.16 | 1.9200e+00 | ▆▆▇▁▁ |
skim(london_house_prices_2019_out_of_sample)| Name | london_house_prices_2019_… |
| Number of rows | 1992 |
| Number of columns | 39 |
| _______________________ | |
| Column type frequency: | |
| character | 1 |
| Date | 1 |
| factor | 11 |
| logical | 6 |
| numeric | 20 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| district | 0 | 1 | 5 | 22 | 0 | 32 | 0 |
Variable type: Date
| skim_variable | n_missing | complete_rate | min | max | median | n_unique |
|---|---|---|---|---|---|---|
| date | 1992 | 0 | NA | NA | NA | 0 |
Variable type: factor
| skim_variable | n_missing | complete_rate | ordered | n_unique | top_counts |
|---|---|---|---|---|---|
| property_type | 0 | 1.00 | FALSE | 4 | F: 778, T: 688, S: 395, D: 131 |
| whether_old_or_new | 0 | 1.00 | FALSE | 2 | N: 1991, Y: 1 |
| freehold_or_leasehold | 0 | 1.00 | FALSE | 2 | F: 1209, L: 783 |
| town | 1909 | 0.04 | FALSE | 54 | CHE: 7, WAL: 7, BLA: 3, FOR: 3 |
| postcode_short | 0 | 1.00 | FALSE | 221 | SW1: 40, CR0: 39, E17: 37, SW1: 28 |
| current_energy_rating | 0 | 1.00 | FALSE | 6 | D: 979, C: 511, E: 398, F: 52 |
| windows_energy_eff | 0 | 1.00 | FALSE | 5 | Ave: 1117, Goo: 470, Ver: 232, Poo: 172 |
| tenure | 0 | 1.00 | FALSE | 3 | own: 1638, ren: 319, ren: 35 |
| nearest_station | 0 | 1.00 | FALSE | 493 | orp: 22, sid: 22, rom: 20, bex: 17 |
| water_company | 0 | 1.00 | FALSE | 4 | Tha: 1491, Aff: 236, Ess: 148, SES: 117 |
| type_of_closest_station | 0 | 1.00 | FALSE | 3 | rai: 969, tub: 659, lig: 364 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| postcode | 1992 | 0 | NaN | : |
| address1 | 1992 | 0 | NaN | : |
| address2 | 1992 | 0 | NaN | : |
| address3 | 1992 | 0 | NaN | : |
| local_aut | 1992 | 0 | NaN | : |
| county | 1992 | 0 | NaN | : |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| ID | 0 | 1 | 7966.98 | 4589.37 | 1.00 | 3955.00 | 7914.00 | 11825.00 | 1.5997e+04 | ▇▇▇▇▇ |
| number_habitable_rooms | 0 | 1 | 4.38 | 1.76 | 1.00 | 3.00 | 4.00 | 5.00 | 1.2000e+01 | ▇▇▃▁▁ |
| co2_emissions_potential | 0 | 1 | 2.30 | 1.55 | 0.10 | 1.30 | 1.90 | 2.80 | 1.6000e+01 | ▇▂▁▁▁ |
| energy_consumption_potential | 0 | 1 | 143.91 | 79.13 | 1.00 | 92.00 | 123.00 | 169.25 | 7.3200e+02 | ▇▃▁▁▁ |
| population | 0 | 1 | 83.06 | 44.35 | 4.00 | 52.00 | 78.00 | 107.00 | 6.2000e+02 | ▇▂▁▁▁ |
| altitude | 0 | 1 | 36.77 | 26.08 | 1.00 | 17.00 | 31.00 | 50.00 | 1.7500e+02 | ▇▅▁▁▁ |
| asking_price | 0 | 1 | 627401.61 | 677878.92 | 83000.00 | 350000.00 | 474000.00 | 670000.00 | 1.6513e+07 | ▇▁▁▁▁ |
| total_floor_area | 0 | 1 | 0.89 | 0.46 | 0.23 | 0.59 | 0.77 | 1.04 | 4.3900e+00 | ▇▂▁▁▁ |
| co2_emissions_current | 0 | 1 | 0.86 | 0.51 | 0.10 | 0.53 | 0.75 | 1.05 | 4.3500e+00 | ▇▃▁▁▁ |
| energy_consumption_current | 0 | 1 | 0.94 | 0.34 | 0.22 | 0.71 | 0.89 | 1.10 | 3.0800e+00 | ▅▇▁▁▁ |
| latitude | 0 | 1 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.0000e+00 | ▂▇▇▇▂ |
| longitude | 0 | 1 | -0.56 | 0.83 | -2.47 | -1.08 | -0.58 | 0.00 | 1.3800e+00 | ▂▆▇▅▂ |
| london_zone | 0 | 1 | 0.94 | 0.35 | 0.25 | 0.74 | 0.99 | 1.24 | 1.4800e+00 | ▁▇▁▅▆ |
| average_income | 0 | 1 | 0.99 | 0.15 | 0.65 | 0.89 | 0.97 | 1.08 | 1.5200e+00 | ▃▇▆▂▁ |
| num_light_rail_lines | 0 | 1 | 0.47 | 0.88 | 0.00 | 0.00 | 0.00 | 0.00 | 2.1200e+00 | ▇▁▁▁▂ |
| num_rail_lines | 0 | 1 | 0.76 | 0.65 | 0.00 | 0.00 | 1.29 | 1.29 | 2.5800e+00 | ▆▁▇▁▁ |
| num_tube_lines | 0 | 1 | 0.51 | 0.86 | 0.00 | 0.00 | 0.00 | 1.15 | 6.9200e+00 | ▇▁▁▁▁ |
| distance_to_station | 0 | 1 | 0.86 | 0.51 | 0.01 | 0.50 | 0.76 | 1.10 | 4.2000e+00 | ▇▅▁▁▁ |
| school_number | 0 | 1 | 0.97 | 0.24 | 0.57 | 0.85 | 0.93 | 1.10 | 1.5200e+00 | ▃▇▆▃▂ |
| crime_number | 0 | 1 | 0.95 | 0.30 | 0.45 | 0.67 | 0.98 | 1.14 | 1.9200e+00 | ▆▇▇▁▁ |
From the skim result, we can see that although some of the variables have high complete rate in training set, the corresponding variables in out-of-sample set is close to 0. For example, post_code is complete in training set but empty in out-of-sample set. This means that in the following model building, this variable should not be used for training purpose, alternatively, postcode_short should be used since it has high complete rate in both training and out-of-sample sets.
set.seed(1) # to return the same result in the following chunks
#let's do the initial split
train_test_split <- initial_split(london_house_prices_2019_training, prop = 0.75) #training set contains 75% of the data
# Create the training dataset
train_data <- training(train_test_split)
test_data <- testing(train_test_split)3 Visualize data
Visualize and examine the data. What plots could be useful here? What do you learn from these visualizations?
ggplot(london_house_prices_2019_training, aes(x=price)) +
geom_density() +
theme_minimal() +
labs(title = "Price is extremely right-skewed",
subtitle = "Density plot of price",
x = "Price",
y = "Density")
We can see from the density plot that the price is mostly concentrated at low value with high-value outliers. Therefore, log transformation will be used to decrease the impact of high-value outliers, and thus making it easier to visualize the distribution of prices.
p1 <- ggplot(london_house_prices_2019_training, aes(x=log(price), fill = property_type)) +
geom_boxplot() +
theme_minimal() +
labs(title = "Outliers exist mostly on right tail for most property types with exception of type F",
subtitle = "Boxplot of log house prices",
x="Log price",
fill = "Property type")
p1
From the boxplot we can see that different property types have different distribution of log price according to the illustration above. We can see that F, which stands for flat, has relatively low prices compared to other types. It also has more lower-end outliers, meaning that some flats (e.g., basement flat) have significantly low prices. Type D, which stands for detached, have on average higher prices than other property types. This is intuitively the case in that detached houses are higher class and more luxurious in general. We can also see that S, which stands for semi-detached houses, has the smallest quantile range.
house_training <- london_house_prices_2019_training %>%
mutate(yrmon= as.yearmon(date)) %>%
group_by(nearest_station) %>%
summarise(median_price = median(price)) %>%
slice_max(median_price, n = 10) %>%
mutate( ToHighlight = ifelse( nearest_station == "hyde park corner"|nearest_station == "knightsbridge", "yes", "no" ) )
p2 <- ggplot(house_training, aes(x = reorder(nearest_station, -median_price), y = median_price, fill = ToHighlight))+
geom_bar(stat="identity")+
scale_fill_manual( values = c( "yes"="tomato", "no"="gray" ), guide = FALSE ) +
theme_minimal() +
theme(axis.text.x=element_text(angle=45, hjust=1)) +
labs(title = "House price near Hyde Park Corner and Knightsbridge are the highest",
subtitle = "Top 10 stations with highest median house price",
x = "Nearest station",
y = "Median house price")
p2
Since average housing price is sensitive to outliers, we use median to represent the housing price level near each station. Looking closely, we can see that the first 6 stations are all a stone’s throw to parks (incl., Hyde Park, Green Park, Regent’s Park). We can also see that Hyde Park Corner station has the highest median price, this is probably because of i) the nice environment surrounded by Hyde Park and Buckingham Palace; ii) prime location which is 23 minutes from the central business district (City of London). The second richest neighborhood is Knightsbridge, which is 7 minutes walk from Hyde Park Corner station. Houses around these two stations seem to be significantly higher than other 8 stations in the top 10 stations with highest median house prices.
london_wards_sf <- read_sf(here("data/London-wards-2018_ESRI/London_Ward.shp"))
city_of_london_wgs84 <- london_wards_sf %>%
st_transform(4326)
price_district <- london_house_prices_2019_training %>%
group_by(district) %>%
summarise(median_price = median(price))
price_district[price_district == "City and County of the City of London"] <- "City of London"
price_district[price_district == "Westminster"] <- "City of Westminster"
price_wards <- city_of_london_wgs84 %>%
left_join(price_district, by=c("DISTRICT" = "district"))
tmap::tmap_mode("view")
tmap::tm_shape(price_wards) +
tm_polygons("median_price",
palette = "Reds",
title='Median housing price among districts') +
tm_layout(title= 'Pricy houses are centered around central-west London',
title.position = c('right', 'top'),
legend.position = c("left","bottom"))From the graph above we can see that the most pricy houses are concentrated around central and west London. This is the case because west London has more advanced infrastructures, parks and mansions that attract the middle to upper class investors. Note that our dataset does not include any observations in the district of the City of London. Therefore, the model developed in the following is not used to predict prices in the City of London.
Estimate a correlation table between prices and other continuous variables. What do you glean from the correlation table?
# produce a correlation table using GGally::ggcor()
library("GGally")
london_house_prices_2019_training %>%
select(-ID) %>% #keep Y variable last
ggcorr(method = c("pairwise", "pearson"), layout.exp = 2,label_round=2, label = TRUE,label_size = 2,hjust = 1,nbreaks = 5,size = 2,angle = -20)+
labs(title = "Price has highest correlation with total floor area",
subtitle ="Correlation table of numeric variables")
Correlation matrix between pairs attributed in Table 1 is presented. The coefficient ranges between -1 and 1, where 1 means two attributes have strong and positive association. In contrast, -1 means two attributes have strong and negative associations. And 0 means the pair is perfectly uncorrelated. As shown in Fig. 2, attribute ‘price’ has a strong positive relationship with attributes ‘total floor area’, ‘number of habitable rooms’, ‘co2 emissions potential’, ‘co2 emissions current’, ‘average income’ and ‘number of tube lines’. In contrast, the attribute ‘Price’ has a strong negative relationship with attributes ‘energy consumption potential’, ‘London zone’. Intuitively, the bigger the house is, the higher the price. This can explain the positive correlation between ‘total floor area’ and ‘price’, ‘number of habitable rooms’ and ‘price’. And the richer the neighborhood, the higher the price of the house. This can explain the positive correlation between ‘average income’ and ‘price’. The more central the location is, the higher the price. This can explain the negative correlation between ‘London zone’ and ‘price’. Since energy efficiency is important for long term investment of a house, the lower the energy consumption, the higher the price. This can explain the negative correlation between ‘energy consumption potential’ and ‘price’. Note that unlike ‘energy consumption current’, which has a high negative correlation with ‘price’, there exhibits a high positive correlation between ‘co2 emission current’ and ‘price’. This is the case because co2 emission, unlike energy consumption, is not penalized in monetary terms for private building owners and therefore is not a critical consideration for investors. And co2 emission is highly correlated with how big the house is. Therefore, the higher the co2 emission current, the higher the price.
4 Fit a linear regression model
To help you get started I build a linear regression model below. I chose a subset of the features with no particular goal. You can (and should) add more variables and/or choose variable selection methods if you want.
Note that among these attributes that are highly correlated with attribute ‘Price’, some pairs of them are highly correlated between themselves (e.g., ‘total floor area’ and ‘co2 emission current’, ‘total floor area’ and ‘number of habitable rooms’, ‘co2 emission current’ and ‘co2 emission potential’). In the following linear regression model training steps, only one attribute from each pair should be used as input for each model. This is due to the multicollinearity problem, which can negatively impact the result of linear regression. This is because the key goal of linear regression is to isolate the relationship between each independent variable (e.g., ‘total floor area’) and the dependent variable (i.e., price). However, when independent variables are correlated (e.g., ‘total floor area’ and ‘co2 emission current’), it means change in ‘total floor area’ shifts ‘co2 emission current’. As a result, the model is difficult to estimate the relationship between each independent variable and the dependent variable since these independent variables tend to change together. Therefore, only one attribute of each pair will be selected as the input into the linear regression model. In the following, I will select ‘total floor area’ as input to avoid multicollinearity.
#Define control variables
control <- trainControl (
method="cv",
number=5,
verboseIter=TRUE) #by setting this to true the model will report its progress after each estimation
#we are going to train the model and report the results using k-fold cross validation
model1_lm<-train(
price ~ latitude+longitude+london_zone*total_floor_area+crime_number+school_number+distance_to_station+population+property_type,
train_data,
method = "lm",
trControl = control,
metric = "RMSE")## + Fold1: intercept=TRUE
## - Fold1: intercept=TRUE
## + Fold2: intercept=TRUE
## - Fold2: intercept=TRUE
## + Fold3: intercept=TRUE
## - Fold3: intercept=TRUE
## + Fold4: intercept=TRUE
## - Fold4: intercept=TRUE
## + Fold5: intercept=TRUE
## - Fold5: intercept=TRUE
## Aggregating results
## Fitting final model on full training set# summary of the results
summary(model1_lm)##
## Call:
## lm(formula = .outcome ~ ., data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1956064 -117359 -7756 97398 5812254
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.118e+07 1.836e+06 -6.089 1.17e-09 ***
## latitude 1.104e+07 1.840e+06 6.002 2.01e-09 ***
## longitude -5.210e+04 3.414e+03 -15.261 < 2e-16 ***
## london_zone 4.753e+05 1.875e+04 25.345 < 2e-16 ***
## total_floor_area 1.710e+06 1.632e+04 104.743 < 2e-16 ***
## crime_number -9.002e+04 1.417e+04 -6.353 2.20e-10 ***
## school_number -6.931e+04 1.548e+04 -4.476 7.68e-06 ***
## distance_to_station -2.823e+04 5.854e+03 -4.821 1.45e-06 ***
## population -8.848e+02 6.495e+01 -13.623 < 2e-16 ***
## property_typeF -1.325e+05 1.550e+04 -8.544 < 2e-16 ***
## property_typeS -1.416e+05 1.377e+04 -10.277 < 2e-16 ***
## property_typeT -1.766e+05 1.409e+04 -12.531 < 2e-16 ***
## `london_zone:total_floor_area` -1.048e+06 1.761e+04 -59.525 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 282900 on 10429 degrees of freedom
## Multiple R-squared: 0.71, Adjusted R-squared: 0.7097
## F-statistic: 2128 on 12 and 10429 DF, p-value: < 2.2e-16# variable importance for LR
importance <- varImp(model1_lm, scale=TRUE)
plot(importance)
4.1 Predict the values in testing and out of sample data
Below I use the predict function to test the performance of the model in testing data and summarize the performance of the linear regression model. How can you measure the quality of your predictions?
set.seed(1) # to return the same result in the following chunks
# We can predict the testing values
predictions <- predict(model1_lm,test_data)
lr_results<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
lr_results ## RMSE Rsquare
## 1 291282.9 0.6657187#We can predict prices for out of sample data the same way
predictions_oos <- predict(model1_lm,london_house_prices_2019_out_of_sample)We can see from the result that Root Mean Square Error (RMSE) is the loss function we try to minimize. Rsquared reached 66.5%, meaning that 66.5% of the variance in the dependent variable price can be explained by the chosen independent variables collectively.
5 Fit a tree model
Next I fit a tree model using the same subset of features. Again you can (and should) add more variables and tune the parameter of your tree to find a better fit.
Compare the performance of the linear regression model with the tree model; which one performs better? Why do you think that is the case?
set.seed(1) # to return the same result in the following chunks
colnames_out_of_sample <- colnames(london_house_prices_2019_out_of_sample)
# pre-process train data for better tree prediction
train_data_tree <- train_data %>%
select(price, colnames_out_of_sample[colnames_out_of_sample != "asking_price"], -date, -address1,-address2,-address3, -town, -postcode, -ID, -local_aut, -county) %>%
na.omit()
trctrl <- trainControl(method = "cv",
number = 5,
classProbs=TRUE)
Grid <- expand.grid(cp = 0.01)
model2_tree <- train(
price ~ latitude+longitude+london_zone+total_floor_area+crime_number+school_number+distance_to_station+property_type,
train_data_tree,
method = "rpart",
metric="RMSE",
trControl=trctrl,
tuneGrid=Grid)
#You can view how the tree performs
model2_tree$results## cp RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 0.01 294609.8 0.6900071 167609.5 25274.21 0.04352829 9144.067#You can view the final tree
rpart.plot(model2_tree$finalModel)
#you can also visualize the variable importance
importance <- varImp(model2_tree, scale=TRUE)
plot(importance)
test_data_tree <- test_data %>%
select(price, colnames_out_of_sample[colnames_out_of_sample != "asking_price"], -date, -address1,-address2,-address3, -town, -postcode, -ID, -local_aut, -county) %>%
na.omit()
predictions <- predict(model2_tree,test_data_tree)
tree_results<-data.frame( RMSE = RMSE(predictions, test_data_tree$price),
Rsquare = R2(predictions, test_data_tree$price))
tree_results ## RMSE Rsquare
## 1 298042.6 0.6519898#We can predict prices for out of sample data the same way
predictions_oos <- predict(model2_tree,london_house_prices_2019_out_of_sample)
predictions_oos## 1 2 3 4 5 6 7 8
## 683984.8 356419.4 356419.4 456465.4 715016.5 456465.4 683984.8 456465.4
## 9 10 11 12 13 14 15 16
## 356419.4 456465.4 456465.4 1368039.7 356419.4 683984.8 356419.4 456465.4
## 17 18 19 20 21 22 23 24
## 356419.4 356419.4 1118283.2 715016.5 456465.4 356419.4 356419.4 1368039.7
## 25 26 27 28 29 30 31 32
## 356419.4 683984.8 356419.4 456465.4 715016.5 683984.8 356419.4 456465.4
## 33 34 35 36 37 38 39 40
## 715016.5 683984.8 683984.8 356419.4 356419.4 356419.4 683984.8 456465.4
## 41 42 43 44 45 46 47 48
## 356419.4 683984.8 683984.8 356419.4 456465.4 356419.4 356419.4 456465.4
## 49 50 51 52 53 54 55 56
## 356419.4 356419.4 456465.4 456465.4 356419.4 356419.4 356419.4 683984.8
## 57 58 59 60 61 62 63 64
## 456465.4 356419.4 683984.8 356419.4 2014519.0 356419.4 356419.4 683984.8
## 65 66 67 68 69 70 71 72
## 356419.4 356419.4 456465.4 1240699.1 683984.8 356419.4 356419.4 456465.4
## 73 74 75 76 77 78 79 80
## 456465.4 456465.4 456465.4 356419.4 356419.4 356419.4 715016.5 356419.4
## 81 82 83 84 85 86 87 88
## 356419.4 456465.4 456465.4 356419.4 2014519.0 3816872.0 356419.4 356419.4
## 89 90 91 92 93 94 95 96
## 683984.8 456465.4 356419.4 683984.8 356419.4 2014519.0 356419.4 356419.4
## 97 98 99 100 101 102 103 104
## 356419.4 356419.4 356419.4 356419.4 356419.4 356419.4 683984.8 356419.4
## 105 106 107 108 109 110 111 112
## 356419.4 356419.4 356419.4 683984.8 456465.4 356419.4 456465.4 456465.4
## 113 114 115 116 117 118 119 120
## 456465.4 356419.4 356419.4 2014519.0 456465.4 683984.8 356419.4 456465.4
## 121 122 123 124 125 126 127 128
## 456465.4 456465.4 456465.4 715016.5 356419.4 456465.4 683984.8 683984.8
## 129 130 131 132 133 134 135 136
## 356419.4 356419.4 356419.4 356419.4 683984.8 356419.4 356419.4 1240699.1
## 137 138 139 140 141 142 143 144
## 356419.4 456465.4 683984.8 715016.5 356419.4 683984.8 356419.4 683984.8
## 145 146 147 148 149 150 151 152
## 715016.5 1368039.7 356419.4 356419.4 356419.4 456465.4 683984.8 456465.4
## 153 154 155 156 157 158 159 160
## 356419.4 356419.4 456465.4 356419.4 456465.4 683984.8 356419.4 715016.5
## 161 162 163 164 165 166 167 168
## 456465.4 356419.4 356419.4 456465.4 456465.4 1368039.7 1240699.1 683984.8
## 169 170 171 172 173 174 175 176
## 456465.4 356419.4 456465.4 2014519.0 1118283.2 356419.4 356419.4 2014519.0
## 177 178 179 180 181 182 183 184
## 356419.4 1240699.1 356419.4 456465.4 456465.4 456465.4 1368039.7 456465.4
## 185 186 187 188 189 190 191 192
## 456465.4 456465.4 1240699.1 456465.4 356419.4 456465.4 456465.4 456465.4
## 193 194 195 196 197 198 199 200
## 356419.4 356419.4 456465.4 356419.4 356419.4 683984.8 456465.4 356419.4
## 201 202 203 204 205 206 207 208
## 456465.4 683984.8 456465.4 356419.4 456465.4 356419.4 456465.4 456465.4
## 209 210 211 212 213 214 215 216
## 356419.4 356419.4 356419.4 456465.4 683984.8 456465.4 456465.4 456465.4
## 217 218 219 220 221 222 223 224
## 356419.4 456465.4 356419.4 715016.5 356419.4 356419.4 356419.4 715016.5
## 225 226 227 228 229 230 231 232
## 356419.4 456465.4 456465.4 456465.4 356419.4 715016.5 456465.4 456465.4
## 233 234 235 236 237 238 239 240
## 356419.4 356419.4 1240699.1 1118283.2 456465.4 356419.4 683984.8 356419.4
## 241 242 243 244 245 246 247 248
## 456465.4 456465.4 456465.4 715016.5 683984.8 356419.4 356419.4 456465.4
## 249 250 251 252 253 254 255 256
## 3399206.7 1240699.1 456465.4 1240699.1 683984.8 456465.4 683984.8 456465.4
## 257 258 259 260 261 262 263 264
## 456465.4 2014519.0 356419.4 715016.5 1118283.2 683984.8 715016.5 683984.8
## 265 266 267 268 269 270 271 272
## 683984.8 356419.4 356419.4 356419.4 1368039.7 715016.5 683984.8 456465.4
## 273 274 275 276 277 278 279 280
## 356419.4 356419.4 356419.4 1118283.2 456465.4 456465.4 1240699.1 356419.4
## 281 282 283 284 285 286 287 288
## 456465.4 356419.4 456465.4 456465.4 1118283.2 356419.4 683984.8 456465.4
## 289 290 291 292 293 294 295 296
## 356419.4 1368039.7 356419.4 356419.4 1118283.2 683984.8 1240699.1 356419.4
## 297 298 299 300 301 302 303 304
## 356419.4 356419.4 456465.4 356419.4 3816872.0 683984.8 356419.4 456465.4
## 305 306 307 308 309 310 311 312
## 1118283.2 456465.4 456465.4 456465.4 356419.4 456465.4 683984.8 456465.4
## 313 314 315 316 317 318 319 320
## 456465.4 356419.4 683984.8 683984.8 456465.4 1118283.2 456465.4 356419.4
## 321 322 323 324 325 326 327 328
## 715016.5 356419.4 456465.4 356419.4 356419.4 683984.8 356419.4 356419.4
## 329 330 331 332 333 334 335 336
## 683984.8 683984.8 356419.4 456465.4 356419.4 683984.8 356419.4 456465.4
## 337 338 339 340 341 342 343 344
## 356419.4 683984.8 356419.4 456465.4 356419.4 356419.4 456465.4 2014519.0
## 345 346 347 348 349 350 351 352
## 683984.8 1240699.1 456465.4 715016.5 356419.4 356419.4 683984.8 683984.8
## 353 354 355 356 357 358 359 360
## 356419.4 456465.4 356419.4 456465.4 456465.4 356419.4 356419.4 683984.8
## 361 362 363 364 365 366 367 368
## 683984.8 456465.4 356419.4 356419.4 456465.4 356419.4 456465.4 715016.5
## 369 370 371 372 373 374 375 376
## 683984.8 456465.4 683984.8 683984.8 683984.8 356419.4 683984.8 456465.4
## 377 378 379 380 381 382 383 384
## 456465.4 683984.8 456465.4 456465.4 456465.4 356419.4 356419.4 2014519.0
## 385 386 387 388 389 390 391 392
## 356419.4 356419.4 683984.8 356419.4 715016.5 683984.8 356419.4 456465.4
## 393 394 395 396 397 398 399 400
## 356419.4 683984.8 356419.4 356419.4 356419.4 456465.4 356419.4 356419.4
## 401 402 403 404 405 406 407 408
## 715016.5 1118283.2 683984.8 1118283.2 683984.8 456465.4 356419.4 456465.4
## 409 410 411 412 413 414 415 416
## 715016.5 356419.4 683984.8 356419.4 683984.8 356419.4 356419.4 356419.4
## 417 418 419 420 421 422 423 424
## 456465.4 456465.4 456465.4 356419.4 683984.8 683984.8 356419.4 456465.4
## 425 426 427 428 429 430 431 432
## 456465.4 456465.4 456465.4 356419.4 683984.8 456465.4 456465.4 683984.8
## 433 434 435 436 437 438 439 440
## 356419.4 1368039.7 456465.4 356419.4 715016.5 715016.5 356419.4 356419.4
## 441 442 443 444 445 446 447 448
## 683984.8 356419.4 715016.5 356419.4 356419.4 356419.4 683984.8 683984.8
## 449 450 451 452 453 454 455 456
## 683984.8 456465.4 683984.8 456465.4 456465.4 1118283.2 456465.4 356419.4
## 457 458 459 460 461 462 463 464
## 2014519.0 356419.4 456465.4 2014519.0 2014519.0 356419.4 356419.4 356419.4
## 465 466 467 468 469 470 471 472
## 683984.8 456465.4 456465.4 456465.4 715016.5 683984.8 456465.4 356419.4
## 473 474 475 476 477 478 479 480
## 3816872.0 456465.4 356419.4 456465.4 715016.5 683984.8 356419.4 683984.8
## 481 482 483 484 485 486 487 488
## 356419.4 1118283.2 356419.4 356419.4 456465.4 456465.4 1118283.2 356419.4
## 489 490 491 492 493 494 495 496
## 356419.4 2014519.0 1368039.7 356419.4 356419.4 683984.8 456465.4 683984.8
## 497 498 499 500 501 502 503 504
## 356419.4 356419.4 1368039.7 456465.4 456465.4 356419.4 715016.5 456465.4
## 505 506 507 508 509 510 511 512
## 456465.4 356419.4 456465.4 683984.8 356419.4 456465.4 356419.4 715016.5
## 513 514 515 516 517 518 519 520
## 1240699.1 456465.4 715016.5 456465.4 456465.4 356419.4 456465.4 683984.8
## 521 522 523 524 525 526 527 528
## 356419.4 456465.4 356419.4 356419.4 683984.8 356419.4 3816872.0 356419.4
## 529 530 531 532 533 534 535 536
## 715016.5 683984.8 683984.8 456465.4 456465.4 456465.4 683984.8 715016.5
## 537 538 539 540 541 542 543 544
## 456465.4 1240699.1 356419.4 356419.4 456465.4 1240699.1 1118283.2 456465.4
## 545 546 547 548 549 550 551 552
## 456465.4 456465.4 683984.8 356419.4 683984.8 1118283.2 456465.4 2014519.0
## 553 554 555 556 557 558 559 560
## 356419.4 1240699.1 356419.4 1118283.2 356419.4 683984.8 456465.4 356419.4
## 561 562 563 564 565 566 567 568
## 356419.4 356419.4 1368039.7 456465.4 356419.4 1240699.1 1118283.2 456465.4
## 569 570 571 572 573 574 575 576
## 356419.4 456465.4 456465.4 456465.4 683984.8 1240699.1 356419.4 356419.4
## 577 578 579 580 581 582 583 584
## 356419.4 683984.8 456465.4 456465.4 356419.4 456465.4 683984.8 356419.4
## 585 586 587 588 589 590 591 592
## 1240699.1 356419.4 683984.8 356419.4 356419.4 1240699.1 456465.4 356419.4
## 593 594 595 596 597 598 599 600
## 1240699.1 356419.4 456465.4 683984.8 456465.4 456465.4 683984.8 456465.4
## 601 602 603 604 605 606 607 608
## 356419.4 683984.8 456465.4 356419.4 683984.8 2014519.0 683984.8 456465.4
## 609 610 611 612 613 614 615 616
## 683984.8 456465.4 456465.4 456465.4 715016.5 356419.4 1118283.2 356419.4
## 617 618 619 620 621 622 623 624
## 356419.4 456465.4 356419.4 456465.4 715016.5 456465.4 456465.4 683984.8
## 625 626 627 628 629 630 631 632
## 456465.4 356419.4 356419.4 456465.4 456465.4 456465.4 456465.4 356419.4
## 633 634 635 636 637 638 639 640
## 456465.4 1368039.7 456465.4 356419.4 356419.4 356419.4 683984.8 683984.8
## 641 642 643 644 645 646 647 648
## 456465.4 356419.4 356419.4 456465.4 715016.5 356419.4 456465.4 356419.4
## 649 650 651 652 653 654 655 656
## 2014519.0 1118283.2 456465.4 356419.4 356419.4 456465.4 456465.4 356419.4
## 657 658 659 660 661 662 663 664
## 1240699.1 683984.8 356419.4 456465.4 356419.4 1118283.2 356419.4 356419.4
## 665 666 667 668 669 670 671 672
## 683984.8 356419.4 456465.4 2014519.0 456465.4 456465.4 683984.8 715016.5
## 673 674 675 676 677 678 679 680
## 1118283.2 683984.8 456465.4 356419.4 2014519.0 356419.4 356419.4 456465.4
## 681 682 683 684 685 686 687 688
## 683984.8 356419.4 356419.4 456465.4 456465.4 356419.4 356419.4 456465.4
## 689 690 691 692 693 694 695 696
## 456465.4 456465.4 456465.4 356419.4 456465.4 356419.4 456465.4 356419.4
## 697 698 699 700 701 702 703 704
## 683984.8 356419.4 1240699.1 2014519.0 456465.4 456465.4 456465.4 356419.4
## 705 706 707 708 709 710 711 712
## 456465.4 456465.4 356419.4 356419.4 456465.4 683984.8 356419.4 456465.4
## 713 714 715 716 717 718 719 720
## 456465.4 356419.4 456465.4 356419.4 683984.8 456465.4 356419.4 356419.4
## 721 722 723 724 725 726 727 728
## 456465.4 456465.4 356419.4 456465.4 456465.4 456465.4 356419.4 356419.4
## 729 730 731 732 733 734 735 736
## 356419.4 683984.8 356419.4 356419.4 456465.4 356419.4 356419.4 1118283.2
## 737 738 739 740 741 742 743 744
## 456465.4 456465.4 456465.4 356419.4 356419.4 683984.8 356419.4 356419.4
## 745 746 747 748 749 750 751 752
## 356419.4 356419.4 356419.4 456465.4 1118283.2 1240699.1 1240699.1 356419.4
## 753 754 755 756 757 758 759 760
## 456465.4 456465.4 683984.8 456465.4 356419.4 683984.8 356419.4 456465.4
## 761 762 763 764 765 766 767 768
## 683984.8 456465.4 356419.4 356419.4 356419.4 1118283.2 356419.4 683984.8
## 769 770 771 772 773 774 775 776
## 715016.5 456465.4 715016.5 356419.4 356419.4 683984.8 356419.4 456465.4
## 777 778 779 780 781 782 783 784
## 2014519.0 456465.4 456465.4 356419.4 356419.4 356419.4 356419.4 356419.4
## 785 786 787 788 789 790 791 792
## 2014519.0 683984.8 456465.4 456465.4 715016.5 356419.4 456465.4 683984.8
## 793 794 795 796 797 798 799 800
## 683984.8 1240699.1 356419.4 356419.4 356419.4 683984.8 456465.4 356419.4
## 801 802 803 804 805 806 807 808
## 1118283.2 1240699.1 456465.4 456465.4 683984.8 456465.4 1118283.2 683984.8
## 809 810 811 812 813 814 815 816
## 2014519.0 456465.4 715016.5 356419.4 356419.4 356419.4 356419.4 356419.4
## 817 818 819 820 821 822 823 824
## 356419.4 683984.8 683984.8 356419.4 683984.8 2014519.0 356419.4 683984.8
## 825 826 827 828 829 830 831 832
## 683984.8 356419.4 356419.4 356419.4 683984.8 456465.4 356419.4 456465.4
## 833 834 835 836 837 838 839 840
## 356419.4 356419.4 456465.4 456465.4 356419.4 683984.8 356419.4 683984.8
## 841 842 843 844 845 846 847 848
## 456465.4 356419.4 1118283.2 456465.4 683984.8 456465.4 683984.8 683984.8
## 849 850 851 852 853 854 855 856
## 1118283.2 715016.5 715016.5 456465.4 456465.4 683984.8 456465.4 456465.4
## 857 858 859 860 861 862 863 864
## 456465.4 356419.4 683984.8 456465.4 683984.8 456465.4 1240699.1 456465.4
## 865 866 867 868 869 870 871 872
## 456465.4 356419.4 715016.5 1240699.1 356419.4 456465.4 683984.8 456465.4
## 873 874 875 876 877 878 879 880
## 356419.4 456465.4 356419.4 356419.4 356419.4 683984.8 1368039.7 456465.4
## 881 882 883 884 885 886 887 888
## 456465.4 1240699.1 356419.4 356419.4 356419.4 456465.4 356419.4 683984.8
## 889 890 891 892 893 894 895 896
## 456465.4 356419.4 683984.8 456465.4 456465.4 715016.5 456465.4 356419.4
## 897 898 899 900 901 902 903 904
## 356419.4 356419.4 715016.5 683984.8 715016.5 356419.4 683984.8 356419.4
## 905 906 907 908 909 910 911 912
## 356419.4 683984.8 683984.8 356419.4 356419.4 356419.4 456465.4 356419.4
## 913 914 915 916 917 918 919 920
## 456465.4 356419.4 715016.5 356419.4 683984.8 456465.4 683984.8 456465.4
## 921 922 923 924 925 926 927 928
## 456465.4 715016.5 356419.4 456465.4 356419.4 456465.4 456465.4 683984.8
## 929 930 931 932 933 934 935 936
## 456465.4 683984.8 356419.4 356419.4 356419.4 356419.4 356419.4 356419.4
## 937 938 939 940 941 942 943 944
## 456465.4 683984.8 356419.4 683984.8 683984.8 356419.4 456465.4 356419.4
## 945 946 947 948 949 950 951 952
## 356419.4 356419.4 356419.4 683984.8 683984.8 683984.8 356419.4 356419.4
## 953 954 955 956 957 958 959 960
## 683984.8 456465.4 456465.4 356419.4 456465.4 683984.8 456465.4 356419.4
## 961 962 963 964 965 966 967 968
## 456465.4 456465.4 683984.8 456465.4 683984.8 456465.4 456465.4 715016.5
## 969 970 971 972 973 974 975 976
## 1118283.2 1240699.1 356419.4 456465.4 456465.4 456465.4 456465.4 456465.4
## 977 978 979 980 981 982 983 984
## 456465.4 715016.5 456465.4 683984.8 456465.4 356419.4 356419.4 456465.4
## 985 986 987 988 989 990 991 992
## 1240699.1 456465.4 456465.4 683984.8 456465.4 356419.4 715016.5 456465.4
## 993 994 995 996 997 998 999 1000
## 356419.4 1240699.1 356419.4 456465.4 356419.4 356419.4 683984.8 456465.4
## 1001 1002 1003 1004 1005 1006 1007 1008
## 456465.4 683984.8 683984.8 456465.4 456465.4 456465.4 356419.4 356419.4
## 1009 1010 1011 1012 1013 1014 1015 1016
## 356419.4 683984.8 356419.4 456465.4 1240699.1 356419.4 356419.4 456465.4
## 1017 1018 1019 1020 1021 1022 1023 1024
## 356419.4 456465.4 683984.8 356419.4 356419.4 456465.4 456465.4 456465.4
## 1025 1026 1027 1028 1029 1030 1031 1032
## 456465.4 683984.8 1368039.7 356419.4 356419.4 1240699.1 456465.4 1368039.7
## 1033 1034 1035 1036 1037 1038 1039 1040
## 456465.4 356419.4 356419.4 456465.4 456465.4 356419.4 683984.8 356419.4
## 1041 1042 1043 1044 1045 1046 1047 1048
## 356419.4 356419.4 456465.4 356419.4 356419.4 683984.8 356419.4 356419.4
## 1049 1050 1051 1052 1053 1054 1055 1056
## 356419.4 456465.4 683984.8 456465.4 356419.4 715016.5 456465.4 1240699.1
## 1057 1058 1059 1060 1061 1062 1063 1064
## 456465.4 456465.4 2014519.0 456465.4 1240699.1 683984.8 356419.4 456465.4
## 1065 1066 1067 1068 1069 1070 1071 1072
## 715016.5 715016.5 683984.8 2014519.0 356419.4 456465.4 1118283.2 456465.4
## 1073 1074 1075 1076 1077 1078 1079 1080
## 356419.4 456465.4 356419.4 683984.8 456465.4 683984.8 356419.4 356419.4
## 1081 1082 1083 1084 1085 1086 1087 1088
## 356419.4 356419.4 456465.4 456465.4 456465.4 683984.8 456465.4 715016.5
## 1089 1090 1091 1092 1093 1094 1095 1096
## 683984.8 356419.4 456465.4 683984.8 356419.4 356419.4 683984.8 456465.4
## 1097 1098 1099 1100 1101 1102 1103 1104
## 356419.4 456465.4 356419.4 356419.4 456465.4 2014519.0 683984.8 715016.5
## 1105 1106 1107 1108 1109 1110 1111 1112
## 356419.4 456465.4 683984.8 2014519.0 356419.4 683984.8 356419.4 2014519.0
## 1113 1114 1115 1116 1117 1118 1119 1120
## 683984.8 456465.4 356419.4 715016.5 456465.4 683984.8 456465.4 456465.4
## 1121 1122 1123 1124 1125 1126 1127 1128
## 356419.4 456465.4 356419.4 683984.8 683984.8 456465.4 456465.4 456465.4
## 1129 1130 1131 1132 1133 1134 1135 1136
## 456465.4 456465.4 456465.4 456465.4 356419.4 456465.4 1240699.1 683984.8
## 1137 1138 1139 1140 1141 1142 1143 1144
## 356419.4 683984.8 683984.8 356419.4 456465.4 356419.4 356419.4 356419.4
## 1145 1146 1147 1148 1149 1150 1151 1152
## 456465.4 1118283.2 715016.5 683984.8 356419.4 356419.4 356419.4 1240699.1
## 1153 1154 1155 1156 1157 1158 1159 1160
## 356419.4 456465.4 356419.4 1240699.1 456465.4 683984.8 456465.4 356419.4
## 1161 1162 1163 1164 1165 1166 1167 1168
## 683984.8 683984.8 683984.8 683984.8 356419.4 683984.8 356419.4 456465.4
## 1169 1170 1171 1172 1173 1174 1175 1176
## 456465.4 683984.8 715016.5 356419.4 356419.4 356419.4 683984.8 456465.4
## 1177 1178 1179 1180 1181 1182 1183 1184
## 356419.4 1240699.1 456465.4 683984.8 683984.8 456465.4 456465.4 715016.5
## 1185 1186 1187 1188 1189 1190 1191 1192
## 456465.4 356419.4 1368039.7 356419.4 356419.4 356419.4 683984.8 356419.4
## 1193 1194 1195 1196 1197 1198 1199 1200
## 356419.4 1240699.1 456465.4 1368039.7 715016.5 356419.4 456465.4 456465.4
## 1201 1202 1203 1204 1205 1206 1207 1208
## 683984.8 1240699.1 356419.4 456465.4 456465.4 356419.4 683984.8 456465.4
## 1209 1210 1211 1212 1213 1214 1215 1216
## 356419.4 456465.4 356419.4 683984.8 356419.4 683984.8 356419.4 456465.4
## 1217 1218 1219 1220 1221 1222 1223 1224
## 456465.4 715016.5 456465.4 683984.8 456465.4 1240699.1 356419.4 456465.4
## 1225 1226 1227 1228 1229 1230 1231 1232
## 456465.4 356419.4 683984.8 683984.8 356419.4 683984.8 456465.4 456465.4
## 1233 1234 1235 1236 1237 1238 1239 1240
## 356419.4 356419.4 356419.4 683984.8 456465.4 683984.8 456465.4 356419.4
## 1241 1242 1243 1244 1245 1246 1247 1248
## 1240699.1 456465.4 356419.4 683984.8 1240699.1 456465.4 356419.4 683984.8
## 1249 1250 1251 1252 1253 1254 1255 1256
## 683984.8 356419.4 456465.4 456465.4 683984.8 456465.4 356419.4 1118283.2
## 1257 1258 1259 1260 1261 1262 1263 1264
## 456465.4 715016.5 356419.4 456465.4 356419.4 456465.4 456465.4 683984.8
## 1265 1266 1267 1268 1269 1270 1271 1272
## 456465.4 456465.4 356419.4 456465.4 715016.5 356419.4 683984.8 356419.4
## 1273 1274 1275 1276 1277 1278 1279 1280
## 456465.4 456465.4 456465.4 2014519.0 456465.4 456465.4 456465.4 456465.4
## 1281 1282 1283 1284 1285 1286 1287 1288
## 456465.4 356419.4 356419.4 715016.5 356419.4 2014519.0 456465.4 356419.4
## 1289 1290 1291 1292 1293 1294 1295 1296
## 1240699.1 356419.4 356419.4 683984.8 456465.4 1118283.2 683984.8 683984.8
## 1297 1298 1299 1300 1301 1302 1303 1304
## 356419.4 356419.4 683984.8 1240699.1 456465.4 356419.4 456465.4 356419.4
## 1305 1306 1307 1308 1309 1310 1311 1312
## 1240699.1 683984.8 456465.4 356419.4 1240699.1 3816872.0 356419.4 356419.4
## 1313 1314 1315 1316 1317 1318 1319 1320
## 356419.4 1240699.1 356419.4 456465.4 1240699.1 683984.8 356419.4 683984.8
## 1321 1322 1323 1324 1325 1326 1327 1328
## 456465.4 456465.4 356419.4 683984.8 683984.8 356419.4 683984.8 456465.4
## 1329 1330 1331 1332 1333 1334 1335 1336
## 683984.8 456465.4 356419.4 456465.4 356419.4 683984.8 356419.4 456465.4
## 1337 1338 1339 1340 1341 1342 1343 1344
## 356419.4 356419.4 1240699.1 456465.4 715016.5 1240699.1 356419.4 1118283.2
## 1345 1346 1347 1348 1349 1350 1351 1352
## 683984.8 683984.8 715016.5 715016.5 715016.5 356419.4 683984.8 356419.4
## 1353 1354 1355 1356 1357 1358 1359 1360
## 683984.8 356419.4 356419.4 683984.8 456465.4 683984.8 1368039.7 456465.4
## 1361 1362 1363 1364 1365 1366 1367 1368
## 1240699.1 683984.8 456465.4 356419.4 456465.4 356419.4 456465.4 356419.4
## 1369 1370 1371 1372 1373 1374 1375 1376
## 456465.4 683984.8 456465.4 1118283.2 356419.4 356419.4 683984.8 356419.4
## 1377 1378 1379 1380 1381 1382 1383 1384
## 456465.4 356419.4 456465.4 715016.5 356419.4 1368039.7 356419.4 356419.4
## 1385 1386 1387 1388 1389 1390 1391 1392
## 456465.4 356419.4 683984.8 683984.8 356419.4 1240699.1 1368039.7 456465.4
## 1393 1394 1395 1396 1397 1398 1399 1400
## 356419.4 356419.4 683984.8 456465.4 356419.4 456465.4 356419.4 683984.8
## 1401 1402 1403 1404 1405 1406 1407 1408
## 356419.4 456465.4 456465.4 456465.4 683984.8 356419.4 356419.4 715016.5
## 1409 1410 1411 1412 1413 1414 1415 1416
## 456465.4 683984.8 456465.4 356419.4 356419.4 356419.4 1240699.1 456465.4
## 1417 1418 1419 1420 1421 1422 1423 1424
## 356419.4 1240699.1 1368039.7 356419.4 456465.4 456465.4 456465.4 456465.4
## 1425 1426 1427 1428 1429 1430 1431 1432
## 715016.5 456465.4 683984.8 356419.4 356419.4 356419.4 1240699.1 456465.4
## 1433 1434 1435 1436 1437 1438 1439 1440
## 683984.8 456465.4 356419.4 356419.4 356419.4 683984.8 356419.4 683984.8
## 1441 1442 1443 1444 1445 1446 1447 1448
## 683984.8 356419.4 3816872.0 356419.4 456465.4 715016.5 715016.5 356419.4
## 1449 1450 1451 1452 1453 1454 1455 1456
## 356419.4 1240699.1 356419.4 356419.4 356419.4 456465.4 456465.4 456465.4
## 1457 1458 1459 1460 1461 1462 1463 1464
## 356419.4 456465.4 683984.8 356419.4 683984.8 456465.4 356419.4 356419.4
## 1465 1466 1467 1468 1469 1470 1471 1472
## 456465.4 356419.4 683984.8 456465.4 1118283.2 456465.4 2014519.0 683984.8
## 1473 1474 1475 1476 1477 1478 1479 1480
## 3816872.0 456465.4 715016.5 1118283.2 356419.4 456465.4 456465.4 356419.4
## 1481 1482 1483 1484 1485 1486 1487 1488
## 356419.4 356419.4 356419.4 356419.4 715016.5 683984.8 356419.4 715016.5
## 1489 1490 1491 1492 1493 1494 1495 1496
## 356419.4 356419.4 683984.8 356419.4 683984.8 456465.4 683984.8 356419.4
## 1497 1498 1499 1500 1501 1502 1503 1504
## 356419.4 356419.4 456465.4 356419.4 356419.4 456465.4 1240699.1 356419.4
## 1505 1506 1507 1508 1509 1510 1511 1512
## 715016.5 356419.4 356419.4 356419.4 356419.4 715016.5 456465.4 356419.4
## 1513 1514 1515 1516 1517 1518 1519 1520
## 1240699.1 456465.4 683984.8 356419.4 456465.4 456465.4 356419.4 456465.4
## 1521 1522 1523 1524 1525 1526 1527 1528
## 356419.4 1118283.2 683984.8 683984.8 456465.4 683984.8 456465.4 456465.4
## 1529 1530 1531 1532 1533 1534 1535 1536
## 456465.4 1240699.1 683984.8 356419.4 456465.4 2014519.0 456465.4 683984.8
## 1537 1538 1539 1540 1541 1542 1543 1544
## 356419.4 715016.5 683984.8 456465.4 356419.4 356419.4 456465.4 456465.4
## 1545 1546 1547 1548 1549 1550 1551 1552
## 356419.4 356419.4 683984.8 356419.4 456465.4 1118283.2 356419.4 356419.4
## 1553 1554 1555 1556 1557 1558 1559 1560
## 456465.4 356419.4 356419.4 356419.4 715016.5 683984.8 356419.4 2014519.0
## 1561 1562 1563 1564 1565 1566 1567 1568
## 1240699.1 683984.8 2014519.0 356419.4 683984.8 456465.4 356419.4 456465.4
## 1569 1570 1571 1572 1573 1574 1575 1576
## 456465.4 456465.4 456465.4 356419.4 456465.4 456465.4 356419.4 356419.4
## 1577 1578 1579 1580 1581 1582 1583 1584
## 1368039.7 456465.4 456465.4 456465.4 356419.4 715016.5 1240699.1 456465.4
## 1585 1586 1587 1588 1589 1590 1591 1592
## 683984.8 456465.4 356419.4 456465.4 456465.4 7939320.0 356419.4 1240699.1
## 1593 1594 1595 1596 1597 1598 1599 1600
## 456465.4 683984.8 683984.8 683984.8 356419.4 456465.4 456465.4 456465.4
## 1601 1602 1603 1604 1605 1606 1607 1608
## 356419.4 683984.8 356419.4 1240699.1 683984.8 456465.4 456465.4 456465.4
## 1609 1610 1611 1612 1613 1614 1615 1616
## 683984.8 456465.4 683984.8 683984.8 1240699.1 356419.4 356419.4 1368039.7
## 1617 1618 1619 1620 1621 1622 1623 1624
## 356419.4 683984.8 356419.4 456465.4 683984.8 715016.5 1118283.2 356419.4
## 1625 1626 1627 1628 1629 1630 1631 1632
## 356419.4 683984.8 456465.4 456465.4 456465.4 456465.4 456465.4 683984.8
## 1633 1634 1635 1636 1637 1638 1639 1640
## 683984.8 683984.8 356419.4 715016.5 456465.4 683984.8 456465.4 356419.4
## 1641 1642 1643 1644 1645 1646 1647 1648
## 683984.8 715016.5 356419.4 715016.5 456465.4 456465.4 456465.4 356419.4
## 1649 1650 1651 1652 1653 1654 1655 1656
## 683984.8 356419.4 683984.8 683984.8 683984.8 456465.4 356419.4 356419.4
## 1657 1658 1659 1660 1661 1662 1663 1664
## 356419.4 683984.8 456465.4 356419.4 456465.4 456465.4 456465.4 683984.8
## 1665 1666 1667 1668 1669 1670 1671 1672
## 456465.4 356419.4 356419.4 356419.4 356419.4 683984.8 356419.4 456465.4
## 1673 1674 1675 1676 1677 1678 1679 1680
## 356419.4 715016.5 456465.4 715016.5 456465.4 456465.4 1240699.1 356419.4
## 1681 1682 1683 1684 1685 1686 1687 1688
## 456465.4 356419.4 683984.8 356419.4 1240699.1 1118283.2 683984.8 456465.4
## 1689 1690 1691 1692 1693 1694 1695 1696
## 356419.4 456465.4 356419.4 683984.8 456465.4 356419.4 715016.5 356419.4
## 1697 1698 1699 1700 1701 1702 1703 1704
## 2014519.0 356419.4 356419.4 683984.8 2014519.0 456465.4 456465.4 3399206.7
## 1705 1706 1707 1708 1709 1710 1711 1712
## 456465.4 456465.4 456465.4 456465.4 715016.5 456465.4 456465.4 683984.8
## 1713 1714 1715 1716 1717 1718 1719 1720
## 456465.4 356419.4 356419.4 683984.8 456465.4 356419.4 356419.4 456465.4
## 1721 1722 1723 1724 1725 1726 1727 1728
## 456465.4 1118283.2 356419.4 356419.4 456465.4 356419.4 683984.8 456465.4
## 1729 1730 1731 1732 1733 1734 1735 1736
## 1368039.7 456465.4 683984.8 683984.8 456465.4 456465.4 356419.4 356419.4
## 1737 1738 1739 1740 1741 1742 1743 1744
## 456465.4 456465.4 456465.4 456465.4 456465.4 683984.8 456465.4 356419.4
## 1745 1746 1747 1748 1749 1750 1751 1752
## 683984.8 456465.4 356419.4 715016.5 356419.4 1240699.1 456465.4 456465.4
## 1753 1754 1755 1756 1757 1758 1759 1760
## 456465.4 456465.4 356419.4 356419.4 356419.4 456465.4 683984.8 683984.8
## 1761 1762 1763 1764 1765 1766 1767 1768
## 356419.4 356419.4 456465.4 356419.4 356419.4 456465.4 456465.4 456465.4
## 1769 1770 1771 1772 1773 1774 1775 1776
## 356419.4 683984.8 356419.4 356419.4 356419.4 456465.4 1118283.2 456465.4
## 1777 1778 1779 1780 1781 1782 1783 1784
## 356419.4 683984.8 356419.4 356419.4 356419.4 715016.5 356419.4 456465.4
## 1785 1786 1787 1788 1789 1790 1791 1792
## 456465.4 1118283.2 683984.8 456465.4 2014519.0 356419.4 456465.4 456465.4
## 1793 1794 1795 1796 1797 1798 1799 1800
## 456465.4 1118283.2 456465.4 715016.5 456465.4 356419.4 456465.4 683984.8
## 1801 1802 1803 1804 1805 1806 1807 1808
## 456465.4 356419.4 1240699.1 356419.4 456465.4 356419.4 683984.8 456465.4
## 1809 1810 1811 1812 1813 1814 1815 1816
## 456465.4 456465.4 456465.4 456465.4 715016.5 456465.4 456465.4 356419.4
## 1817 1818 1819 1820 1821 1822 1823 1824
## 456465.4 683984.8 356419.4 356419.4 356419.4 356419.4 356419.4 1118283.2
## 1825 1826 1827 1828 1829 1830 1831 1832
## 683984.8 356419.4 456465.4 683984.8 456465.4 356419.4 456465.4 356419.4
## 1833 1834 1835 1836 1837 1838 1839 1840
## 356419.4 1118283.2 715016.5 456465.4 356419.4 456465.4 356419.4 715016.5
## 1841 1842 1843 1844 1845 1846 1847 1848
## 456465.4 456465.4 356419.4 456465.4 683984.8 715016.5 456465.4 1118283.2
## 1849 1850 1851 1852 1853 1854 1855 1856
## 3399206.7 2014519.0 715016.5 1240699.1 356419.4 1240699.1 356419.4 683984.8
## 1857 1858 1859 1860 1861 1862 1863 1864
## 2014519.0 456465.4 356419.4 715016.5 456465.4 356419.4 456465.4 356419.4
## 1865 1866 1867 1868 1869 1870 1871 1872
## 715016.5 715016.5 456465.4 356419.4 356419.4 715016.5 1240699.1 683984.8
## 1873 1874 1875 1876 1877 1878 1879 1880
## 1368039.7 356419.4 356419.4 356419.4 1368039.7 356419.4 456465.4 683984.8
## 1881 1882 1883 1884 1885 1886 1887 1888
## 456465.4 683984.8 356419.4 715016.5 683984.8 1240699.1 683984.8 456465.4
## 1889 1890 1891 1892 1893 1894 1895 1896
## 456465.4 356419.4 683984.8 456465.4 356419.4 356419.4 356419.4 356419.4
## 1897 1898 1899 1900 1901 1902 1903 1904
## 1118283.2 456465.4 456465.4 456465.4 456465.4 456465.4 683984.8 683984.8
## 1905 1906 1907 1908 1909 1910 1911 1912
## 356419.4 683984.8 356419.4 456465.4 456465.4 456465.4 683984.8 356419.4
## 1913 1914 1915 1916 1917 1918 1919 1920
## 356419.4 456465.4 356419.4 715016.5 456465.4 456465.4 456465.4 456465.4
## 1921 1922 1923 1924 1925 1926 1927 1928
## 715016.5 1240699.1 683984.8 1118283.2 356419.4 356419.4 356419.4 456465.4
## 1929 1930 1931 1932 1933 1934 1935 1936
## 356419.4 683984.8 356419.4 456465.4 356419.4 683984.8 456465.4 683984.8
## 1937 1938 1939 1940 1941 1942 1943 1944
## 683984.8 356419.4 456465.4 356419.4 356419.4 356419.4 356419.4 356419.4
## 1945 1946 1947 1948 1949 1950 1951 1952
## 356419.4 1240699.1 356419.4 456465.4 683984.8 683984.8 456465.4 356419.4
## 1953 1954 1955 1956 1957 1958 1959 1960
## 715016.5 683984.8 715016.5 356419.4 456465.4 683984.8 456465.4 356419.4
## 1961 1962 1963 1964 1965 1966 1967 1968
## 356419.4 1118283.2 456465.4 683984.8 1240699.1 356419.4 683984.8 456465.4
## 1969 1970 1971 1972 1973 1974 1975 1976
## 456465.4 356419.4 356419.4 356419.4 456465.4 356419.4 356419.4 356419.4
## 1977 1978 1979 1980 1981 1982 1983 1984
## 356419.4 1118283.2 356419.4 456465.4 683984.8 356419.4 715016.5 456465.4
## 1985 1986 1987 1988 1989 1990 1991 1992
## 715016.5 356419.4 356419.4 456465.4 683984.8 715016.5 715016.5 356419.4We can see from the result that the RMSE reached 294609.8, which is 1.1% higher than the RMSE obtained from the linear regression (LR) model. This is the case probably because the tree model predicts the price range instead of the exact price as LR does. Therefore the error is higher. Rsquared reached 69.0%, which is 2.5% higher than that of LR model. This is the case probably because many relationships between independent variables and dependent variable are non-linear, and LR fails to detect the non-linear relationship. Tree model is able to capture the non-linear relationship. Therefore, Rsquared is higher.
6 Other algorithms
Use at least two other algorithms to predict prices. Don’t forget to tune the parameters of these algorithms. And then compare the performances of your algorithms to linear regression and trees.
set.seed(1)
model3 <- train(
price ~ latitude+longitude+london_zone+total_floor_area+crime_number+school_number+distance_to_station+co2_emissions_current+property_type,
train_data,
method = "knn",
trControl = control,
tuneLength=10
)## + Fold1: k= 5
## - Fold1: k= 5
## + Fold1: k= 7
## - Fold1: k= 7
## + Fold1: k= 9
## - Fold1: k= 9
## + Fold1: k=11
## - Fold1: k=11
## + Fold1: k=13
## - Fold1: k=13
## + Fold1: k=15
## - Fold1: k=15
## + Fold1: k=17
## - Fold1: k=17
## + Fold1: k=19
## - Fold1: k=19
## + Fold1: k=21
## - Fold1: k=21
## + Fold1: k=23
## - Fold1: k=23
## + Fold2: k= 5
## - Fold2: k= 5
## + Fold2: k= 7
## - Fold2: k= 7
## + Fold2: k= 9
## - Fold2: k= 9
## + Fold2: k=11
## - Fold2: k=11
## + Fold2: k=13
## - Fold2: k=13
## + Fold2: k=15
## - Fold2: k=15
## + Fold2: k=17
## - Fold2: k=17
## + Fold2: k=19
## - Fold2: k=19
## + Fold2: k=21
## - Fold2: k=21
## + Fold2: k=23
## - Fold2: k=23
## + Fold3: k= 5
## - Fold3: k= 5
## + Fold3: k= 7
## - Fold3: k= 7
## + Fold3: k= 9
## - Fold3: k= 9
## + Fold3: k=11
## - Fold3: k=11
## + Fold3: k=13
## - Fold3: k=13
## + Fold3: k=15
## - Fold3: k=15
## + Fold3: k=17
## - Fold3: k=17
## + Fold3: k=19
## - Fold3: k=19
## + Fold3: k=21
## - Fold3: k=21
## + Fold3: k=23
## - Fold3: k=23
## + Fold4: k= 5
## - Fold4: k= 5
## + Fold4: k= 7
## - Fold4: k= 7
## + Fold4: k= 9
## - Fold4: k= 9
## + Fold4: k=11
## - Fold4: k=11
## + Fold4: k=13
## - Fold4: k=13
## + Fold4: k=15
## - Fold4: k=15
## + Fold4: k=17
## - Fold4: k=17
## + Fold4: k=19
## - Fold4: k=19
## + Fold4: k=21
## - Fold4: k=21
## + Fold4: k=23
## - Fold4: k=23
## + Fold5: k= 5
## - Fold5: k= 5
## + Fold5: k= 7
## - Fold5: k= 7
## + Fold5: k= 9
## - Fold5: k= 9
## + Fold5: k=11
## - Fold5: k=11
## + Fold5: k=13
## - Fold5: k=13
## + Fold5: k=15
## - Fold5: k=15
## + Fold5: k=17
## - Fold5: k=17
## + Fold5: k=19
## - Fold5: k=19
## + Fold5: k=21
## - Fold5: k=21
## + Fold5: k=23
## - Fold5: k=23
## Aggregating results
## Selecting tuning parameters
## Fitting k = 7 on full training set# view how the tree performs
model3$results## k RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 5 268334.7 0.7419127 122908.1 35002.73 0.04531215 6972.112
## 2 7 264300.5 0.7513824 121081.2 34503.87 0.04418249 6622.188
## 3 9 269796.0 0.7429650 121359.7 31435.99 0.03421529 6380.043
## 4 11 271217.9 0.7405854 121333.5 27911.04 0.02584640 6058.587
## 5 13 271880.1 0.7413560 121822.1 26284.24 0.02129176 6006.266
## 6 15 273521.4 0.7394451 122017.0 26610.99 0.02218062 5647.180
## 7 17 276592.0 0.7346948 122582.0 27617.31 0.02040363 5854.606
## 8 19 278407.9 0.7327221 122754.8 28946.91 0.02280756 6046.439
## 9 21 280663.9 0.7301074 123176.6 30938.59 0.02628859 5946.114
## 10 23 281850.0 0.7293300 123681.0 30807.98 0.02549628 5944.449knn_performance <- data.frame(k=model3$results$k,
RMSE=model3$results$RMSE,
Rsquared=model3$results$Rsquared)
highlight <- knn_performance %>%
filter(k==7)
# RMSE for different K
ggplot(data=knn_performance, aes(x=k, y=RMSE, group=1)) +
geom_line()+
geom_point()+
geom_point(data=highlight,
aes(x=k,y=RMSE),
color='red',
size=3) +
theme_minimal() +
labs(title = "Model achieved lowest RMSE at K=7",
subtitle = "RMSE with different K values",
x = "K",
y = "RMSE")
# Rsquared for different K
ggplot(data=knn_performance, aes(x=k, y=Rsquared, group=1)) +
geom_line()+
geom_point()+
geom_point(data=highlight,
aes(x=k,y=Rsquared),
color='red',
size=3) +
theme_minimal() +
labs(title = "Model achieved highest Rsquared at K=7",
subtitle = "Rsquared with different K values",
x = "K",
y = "Rsquared")
We can see that performance achieved its best at k=7.
model3_knn <- train(
price ~ latitude+longitude+london_zone+total_floor_area+crime_number+school_number+distance_to_station+co2_emissions_current+property_type,
train_data,
method = "knn",
trControl = control,
tuneGrid = expand.grid(k = 7) #optimal K
)## + Fold1: k=7
## - Fold1: k=7
## + Fold2: k=7
## - Fold2: k=7
## + Fold3: k=7
## - Fold3: k=7
## + Fold4: k=7
## - Fold4: k=7
## + Fold5: k=7
## - Fold5: k=7
## Aggregating results
## Fitting final model on full training set#you can also visualize the variable importance
importance <- varImp(model3_knn, scale=TRUE)
plot(importance)
predictions <- predict(model3_knn,test_data)
knn_results<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
knn_results## RMSE Rsquare
## 1 244964.8 0.7634802#We can predict prices for out of sample data the same way
predictions_oos <- predict(model3_knn,london_house_prices_2019_out_of_sample)
predictions_oos## [1] 896682.3 332135.7 441000.0 470000.0 986642.9 525550.0 873857.1
## [8] 399785.7 735428.6 556213.6 399000.0 1000895.3 463000.0 809714.3
## [15] 411857.1 486714.3 331714.3 284428.6 989428.6 1073571.4 523617.9
## [22] 279642.9 267135.7 1295714.3 797071.4 586357.1 263214.3 273857.1
## [29] 782500.0 447142.9 412714.3 464028.6 750857.1 819642.9 629142.9
## [36] 424214.3 844971.4 255857.1 595992.9 655714.3 341428.6 986609.6
## [43] 410571.4 634642.9 424107.1 451422.4 342857.1 567000.0 540642.9
## [50] 305428.6 459285.7 373428.6 429571.4 364035.7 490642.9 646928.6
## [57] 481714.3 360098.4 529857.1 300421.4 3503571.3 237928.6 351571.4
## [64] 686600.0 191071.4 246285.7 763214.3 947857.1 787285.7 345971.1
## [71] 603928.6 561857.1 362071.4 413357.1 250964.3 337850.0 384642.9
## [78] 342928.6 841857.1 356714.3 295285.7 527000.0 421690.6 582285.7
## [85] 1555714.3 2395714.3 293714.3 253685.7 823785.7 426714.3 503957.1
## [92] 1166071.4 413735.4 1556642.9 401814.3 266642.9 354293.3 336857.1
## [99] 318642.9 573178.6 224785.7 370071.4 676071.4 517571.1 290142.9
## [106] 348564.3 243785.7 485500.0 411142.9 441071.4 608571.4 393464.3
## [113] 444071.4 410500.0 370428.6 1628428.6 401571.4 695714.3 217142.9
## [120] 554000.0 705714.3 365785.7 402857.1 702142.9 849514.3 525357.1
## [127] 561857.1 512857.1 451428.6 445785.7 614000.0 454714.3 700214.3
## [134] 423571.4 345421.4 1332500.0 286142.9 403000.0 464642.9 743571.4
## [141] 353285.7 799285.7 225341.4 642427.9 510285.7 891000.0 226202.9
## [148] 301314.3 328000.0 411142.9 567298.6 366714.3 228357.1 330714.1
## [155] 475000.0 248571.4 401428.6 776785.7 431614.3 956714.3 341285.7
## [162] 211428.6 295500.0 360714.3 541857.1 1361428.6 1001428.6 682428.6
## [169] 608425.7 522248.6 423571.4 1775357.1 913357.1 242642.9 378492.9
## [176] 1300285.7 291857.1 798571.4 630000.0 429000.0 396571.4 369928.6
## [183] 1806071.4 458071.4 683142.9 728000.0 1335142.9 506707.1 266142.1
## [190] 463000.0 474428.6 504835.7 312678.3 344285.7 455714.3 231714.3
## [197] 324214.3 673000.0 457857.1 300785.7 526357.1 751785.7 568907.1
## [204] 249285.7 423571.4 323207.1 260142.9 627333.9 415928.6 373714.3
## [211] 357785.7 354428.6 734714.3 488642.9 875214.3 350760.6 443285.7
## [218] 452857.1 300357.1 505857.1 232857.1 310071.4 454371.4 666017.9
## [225] 246642.9 310928.6 677642.9 360285.7 223642.9 1067071.4 446928.6
## [232] 550857.1 380064.3 347214.3 1497857.1 1251847.4 469785.7 262071.4
## [239] 570500.0 281500.0 327571.4 311321.4 355785.7 1812857.1 681357.0
## [246] 413928.6 414714.3 448642.9 6319553.6 923857.1 431785.7 1670714.3
## [253] 529285.7 313285.7 575142.9 535621.4 465714.3 2200168.3 315928.6
## [260] 780357.1 1072500.0 591285.7 560571.4 595285.7 713857.0 604428.6
## [267] 369428.6 351221.9 1348571.4 496142.9 1008214.3 406428.6 297000.0
## [274] 430142.9 401785.6 1567142.9 394857.1 402857.1 2252142.9 371714.1
## [281] 1760607.1 435857.1 565714.3 362428.6 805142.9 384642.9 901428.6
## [288] 417214.3 410947.7 1232857.1 349500.0 372928.6 1876785.7 758571.4
## [295] 911040.9 506428.6 352357.1 646428.6 361714.3 262285.7 3885000.0
## [302] 521428.6 225428.6 552000.0 1130714.3 334357.1 438714.3 349214.3
## [309] 301721.9 808571.4 649428.6 409571.4 424071.4 505500.0 784714.3
## [316] 1018357.1 453571.4 1300428.6 504642.9 393850.0 1109285.7 505357.1
## [323] 261607.1 616400.0 352142.9 1110571.4 286371.4 279785.7 545642.9
## [330] 687142.9 298000.0 394071.4 314778.6 761421.4 264642.9 495671.4
## [337] 371428.6 637142.9 299642.9 404285.7 361357.1 246350.0 486642.9
## [344] 1235500.0 663142.9 1154285.7 407042.9 549071.4 228071.4 338000.0
## [351] 484571.4 502321.4 508000.0 392785.7 494000.0 428571.4 475571.4
## [358] 530357.1 323500.0 485071.4 524285.7 395285.7 211214.3 401427.9
## [365] 590928.6 477231.4 487142.9 576178.6 447571.4 447214.3 684021.4
## [372] 722561.4 659071.4 360421.4 577321.4 341214.3 395214.3 514857.1
## [379] 414571.4 436028.6 457857.1 324500.0 318714.3 1211857.1 353384.1
## [386] 387142.9 526857.1 251214.3 785000.0 505928.6 292928.6 513142.9
## [393] 372928.6 911642.9 319071.4 480721.4 428232.6 585714.3 289000.0
## [400] 307778.6 835568.9 828714.3 462600.1 997928.6 844714.3 504857.1
## [407] 400000.0 363500.0 448928.6 489142.9 603928.6 564650.0 559714.3
## [414] 427214.3 312903.4 359357.1 451071.4 455742.9 551142.9 519571.4
## [421] 1063214.3 463914.3 320000.0 400642.9 524000.0 469285.7 627571.4
## [428] 215971.4 685285.7 393928.6 611357.1 738142.9 224528.6 1812857.1
## [435] 528857.1 338285.7 613785.7 822142.9 527857.1 263571.4 676278.6
## [442] 438598.4 810000.0 564000.0 296285.7 401000.0 653492.9 855785.7
## [449] 720000.0 489364.3 655428.6 596312.9 415760.9 2131428.6 448928.6
## [456] 305857.1 1460625.0 451571.4 335117.7 1549642.9 1418142.9 214142.9
## [463] 230714.3 230000.0 878317.7 529214.3 468584.9 435642.9 892857.1
## [470] 646857.1 327000.0 410714.3 3947142.9 439428.6 311857.1 243000.0
## [477] 587357.1 455000.0 551492.9 739214.3 336321.4 1040000.0 577142.9
## [484] 388571.4 395071.4 457214.3 813214.3 203757.1 407527.0 1754642.9
## [491] 1663571.4 226000.0 203457.1 850814.3 429142.9 664642.9 499999.9
## [498] 417785.7 1302857.1 705350.0 499642.7 392135.7 1155125.0 491142.9
## [505] 322500.0 364035.7 556142.9 457857.1 229557.1 583571.4 357678.6
## [512] 588583.1 1389219.9 544642.9 859285.7 391285.7 543571.3 288916.0
## [519] 515142.9 729857.1 202785.7 471500.0 380064.3 431964.3 574142.9
## [526] 317571.4 4218571.4 227142.9 685357.1 499285.7 503285.7 335714.3
## [533] 712428.6 446285.7 628500.0 716997.4 315757.1 1215642.9 299107.1
## [540] 403571.4 499714.3 618589.3 1141500.0 463214.3 593448.6 370642.9
## [547] 815428.6 389642.9 642642.9 1422000.0 375528.6 1896714.3 455000.0
## [554] 1036421.4 497071.4 1280714.3 364708.3 499428.6 469071.4 217142.9
## [561] 433535.7 357785.7 1169714.3 549428.6 269357.1 1067857.0 642785.7
## [568] 440214.3 267350.0 628571.4 464142.9 478571.4 454285.7 1430000.0
## [575] 403071.4 308135.7 363642.9 926928.6 503000.0 358357.1 322428.6
## [582] 409571.4 658792.1 299428.6 1676850.0 194000.0 853571.4 370928.6
## [589] 283857.1 1127450.0 356464.3 252357.1 974214.3 203357.1 746071.4
## [596] 1234285.7 606971.4 592500.0 463914.3 305142.9 350000.0 483000.0
## [603] 541714.3 267642.9 850785.7 1329857.1 417057.1 778571.4 523857.1
## [610] 619785.7 480714.3 339785.7 449142.9 402857.1 918714.3 280714.3
## [617] 631857.1 468571.4 349528.6 469241.1 744428.6 379571.4 407807.1
## [624] 424285.7 683142.9 242500.0 215270.0 397857.1 412714.3 635285.7
## [631] 453178.6 394828.6 549571.4 1889285.7 598000.0 326285.7 222357.1
## [638] 291308.9 685571.4 627642.9 340571.4 268785.7 279928.6 574285.7
## [645] 961642.9 326300.0 406421.4 469278.6 1019428.6 1353214.3 685498.6
## [652] 403142.9 372928.6 477642.7 381428.6 286071.4 1019071.4 669000.0
## [659] 379142.9 395142.9 465000.0 2014985.7 235035.7 292928.6 962499.9
## [666] 445264.3 455428.6 3284985.7 579857.1 319428.6 834500.0 634107.1
## [673] 1118214.3 687428.6 419500.0 455400.0 1609285.7 734642.9 241535.7
## [680] 701142.9 417642.9 380785.7 499999.9 528428.6 594306.4 391500.0
## [687] 529500.0 385642.9 445214.3 543571.4 512500.0 404350.0 699428.6
## [694] 492755.0 361785.7 261357.1 579642.9 360742.9 2546071.4 1987142.9
## [701] 370428.6 334642.9 452427.7 327142.9 565285.7 487857.1 304278.6
## [708] 208428.6 553500.0 540428.6 265214.3 526857.1 461285.7 408428.6
## [715] 398214.3 216428.6 959928.6 538857.1 406857.1 426642.9 403157.1
## [722] 566428.6 573928.6 403785.7 495071.4 543285.7 311071.4 520000.0
## [729] 573785.7 932371.4 588428.6 375785.7 489600.0 414000.0 539648.7
## [736] 2373428.6 625000.0 603142.9 536000.0 430857.1 267257.1 516428.6
## [743] 309028.6 331928.6 225000.0 437785.7 256500.0 319928.6 1102928.6
## [750] 1224571.4 1934285.7 509928.6 416428.6 518714.3 577428.6 399857.1
## [757] 408642.9 560000.0 369357.1 380428.6 841857.1 426571.4 342000.0
## [764] 346071.4 514142.9 1006857.1 553571.4 598442.9 520714.3 659714.3
## [771] 490000.0 633285.7 371642.9 1088571.4 498571.4 483714.3 2252142.9
## [778] 429571.4 327829.3 589142.9 386428.6 457821.1 578857.1 364928.6
## [785] 2775642.9 566785.7 503142.9 486142.9 510828.6 469785.7 266571.4
## [792] 962142.9 731571.4 1026500.0 354642.9 305928.6 237485.7 685492.9
## [799] 717000.0 399857.1 1165000.0 1228071.4 380357.1 344885.7 580928.6
## [806] 461428.6 1195714.3 458214.3 1727142.9 432785.7 462142.9 237857.1
## [813] 760357.1 421714.3 343285.7 549142.9 609857.1 658142.9 457642.9
## [820] 251428.6 728857.0 2479285.7 235071.4 1463000.0 809642.9 533339.6
## [827] 362500.0 518500.0 563528.6 519357.1 327428.6 469285.7 467535.4
## [834] 297285.7 418571.4 607285.7 207500.0 578571.4 445714.3 481571.4
## [841] 438571.4 235857.1 1187857.1 361928.6 944828.6 424000.0 643782.9
## [848] 460999.7 950785.7 924285.7 1367706.0 505357.1 384642.9 524999.7
## [855] 411428.6 398000.0 525414.3 268999.3 589571.4 314142.9 619992.9
## [862] 551873.6 2175785.7 435378.6 547428.6 393535.7 590714.3 1077171.4
## [869] 365214.3 552857.1 741857.1 425278.4 231928.6 490285.7 374741.3
## [876] 451857.1 381964.3 585714.3 1240471.4 791785.7 477421.4 1065893.0
## [883] 241785.7 234214.3 343428.6 491785.7 461142.9 765285.7 368071.4
## [890] 237500.0 590421.4 413464.3 690857.1 472571.4 705385.7 356214.3
## [897] 326000.0 482421.4 498135.7 523428.6 451514.3 320607.1 928714.3
## [904] 311028.6 563714.3 611142.9 673714.3 248500.0 372142.9 213000.0
## [911] 580857.1 286428.6 368321.4 304642.9 423821.4 309300.0 816664.3
## [918] 328428.6 691857.1 381071.4 404142.9 614127.1 375785.7 352500.0
## [925] 279000.0 601806.4 510468.6 696071.4 406928.6 493428.6 232474.3
## [932] 324927.9 544928.6 363571.4 240992.9 223714.3 333100.0 1390428.6
## [939] 256071.4 806428.6 533321.4 498428.6 328428.6 481000.0 491084.9
## [946] 377142.9 345928.6 406635.7 520428.6 677714.3 221142.9 340035.7
## [953] 853506.4 571714.3 366785.7 352571.4 516000.0 404000.0 497928.6
## [960] 261988.6 546071.4 392142.9 1095357.9 377357.1 718102.9 567298.6
## [967] 384857.1 1620714.3 1839285.7 1300421.4 856071.4 623571.4 339357.1
## [974] 378571.4 392000.0 474642.7 490857.1 744500.0 346857.1 655214.3
## [981] 476428.6 401642.9 307357.1 610155.7 1466428.6 501071.4 520421.4
## [988] 770714.3 343857.1 255342.9 783187.5 450000.0 454000.0 1798571.4
## [995] 504500.0 353264.3 373714.3 304285.7 447514.3 424857.1 438785.7
## [1002] 514500.0 665000.0 540714.3 415285.7 344928.6 318428.6 344135.7
## [1009] 230571.4 543214.3 443357.1 343857.1 908142.9 407500.0 403785.7
## [1016] 544278.6 418064.3 322142.9 426500.0 298564.3 209627.1 408928.6
## [1023] 380928.6 372571.3 554428.6 542064.3 1969739.7 403571.4 336285.7
## [1030] 825912.9 487511.4 819571.4 330571.4 292142.9 475828.6 658285.7
## [1037] 498357.1 246000.0 580785.7 441000.0 259714.3 614142.9 473285.7
## [1044] 257285.7 452000.0 412135.7 216571.4 258500.0 395285.7 360714.3
## [1051] 801350.0 391214.3 379707.1 632642.1 521714.3 1752142.9 377285.7
## [1058] 477428.6 2995000.0 526714.3 1378214.3 688714.3 474285.7 389214.3
## [1065] 784242.9 732071.4 360142.9 1235357.1 241785.7 444578.6 1763785.7
## [1072] 345571.4 359285.7 417619.1 399071.4 931928.6 427428.6 604421.4
## [1079] 276785.7 331714.3 377714.3 430564.3 456357.1 849785.7 364714.3
## [1086] 676785.7 374928.6 853142.9 883857.1 317642.9 721428.6 415028.6
## [1093] 259628.6 386571.4 871428.6 557857.1 389271.4 383928.6 308000.0
## [1100] 196857.1 408785.7 1222178.6 650512.9 604428.6 342214.3 543857.1
## [1107] 482500.0 1941714.3 445371.4 493678.4 324000.0 3922142.9 795950.0
## [1114] 420714.3 347357.1 540142.9 666071.4 513142.9 327600.0 362500.0
## [1121] 393321.4 364142.9 206714.3 651071.4 528285.7 407500.0 413928.6
## [1128] 626714.3 495714.3 380857.1 401135.7 426000.0 220000.0 314714.3
## [1135] 1176471.4 741285.7 517714.3 596142.9 649357.1 327214.3 362285.7
## [1142] 442571.1 666304.9 200500.0 413928.6 953857.1 552714.3 498142.9
## [1149] 196607.1 199714.3 654428.6 1140428.6 337000.0 480785.7 275071.4
## [1156] 2907142.9 400000.0 503928.6 370785.7 298057.1 764285.7 775571.4
## [1163] 606157.1 589000.0 248642.9 504857.1 385900.0 590000.0 539750.0
## [1170] 603214.3 598214.3 325714.3 474600.0 576428.4 686282.9 583428.6
## [1177] 373900.0 994285.7 498071.4 612000.0 543857.1 419642.9 512000.0
## [1184] 548571.4 540642.9 363721.9 1471142.9 333144.1 297642.9 269428.6
## [1191] 515857.1 231857.1 417857.1 1333014.3 392428.6 1497857.1 542142.9
## [1198] 468928.6 472928.6 438857.1 546714.3 1367300.0 460928.6 336428.6
## [1205] 524714.3 209285.7 561142.9 540671.4 587071.4 408928.6 224642.9
## [1212] 768714.3 340500.0 593571.4 224928.6 412285.6 492285.7 635805.7
## [1219] 478928.6 589142.9 543285.7 644285.7 270428.6 448778.6 454500.0
## [1226] 232571.4 845357.1 737500.0 534714.3 687428.6 489333.6 378142.9
## [1233] 406785.7 212571.4 346107.1 1067285.7 381285.7 547142.9 670500.0
## [1240] 447850.0 1176850.0 692064.3 542071.4 488771.4 698714.3 513857.1
## [1247] 327642.7 494285.7 715671.4 306142.9 668118.6 333642.9 1057000.0
## [1254] 294928.6 322664.3 971142.9 408071.4 1284143.6 336785.7 758071.4
## [1261] 237785.7 494071.4 608100.0 484999.7 347071.4 398285.7 441807.1
## [1268] 534142.9 1142489.4 250000.0 655000.0 345142.9 363000.0 587714.3
## [1275] 463500.0 3707142.9 353142.9 757114.3 440500.0 504428.6 375000.0
## [1282] 421142.9 240625.0 548442.0 430714.3 1222178.6 477214.3 638125.7
## [1289] 1076428.6 361857.1 466157.1 621642.9 525785.7 1174172.9 699746.3
## [1296] 522421.3 448999.9 565000.0 540142.9 2555714.3 429357.1 341428.6
## [1303] 474571.4 394428.6 930257.1 515142.9 470000.0 296000.0 1136500.0
## [1310] 5784742.9 360000.0 400942.9 418785.7 1183685.7 427178.6 539642.7
## [1317] 1336428.6 616071.4 359357.1 555142.9 378500.0 398357.1 229198.6
## [1324] 677500.0 875314.3 753000.0 475528.7 580642.9 733714.3 606571.4
## [1331] 273285.7 417142.9 394500.0 870000.0 257714.3 390928.6 355142.9
## [1338] 449071.4 1345785.7 573285.7 499660.7 2382857.1 312964.0 862571.4
## [1345] 636071.4 512500.0 491085.7 486542.9 518000.0 618285.7 830321.4
## [1352] 276571.4 663285.7 371714.3 510341.4 899285.7 418928.6 390714.3
## [1359] 2555000.0 423428.6 1118357.1 571000.0 421285.7 348928.6 374714.3
## [1366] 349500.0 394285.7 255285.7 413250.0 902857.1 419785.7 932857.1
## [1373] 263571.4 239357.1 700000.0 662768.6 376857.1 398857.1 513714.3
## [1380] 529142.9 273714.3 2080142.9 278178.6 385128.6 263232.4 577785.7
## [1387] 871714.3 378928.6 462714.3 1113357.1 1902357.1 432000.0 239000.0
## [1394] 435357.1 379714.3 432571.4 372785.7 556916.0 459785.7 927357.1
## [1401] 360821.4 425642.9 401342.9 475357.1 447514.3 529733.4 224857.1
## [1408] 628492.9 490642.9 946071.4 343642.9 271142.9 292642.9 374857.1
## [1415] 1021857.1 434142.9 373214.3 758928.6 2189285.7 343928.6 436714.3
## [1422] 330400.7 331829.3 431107.1 611785.7 396214.3 727114.3 332142.9
## [1429] 312500.0 411241.3 1340128.6 344514.3 659571.4 361071.4 398714.3
## [1436] 451428.6 286142.9 440571.4 390160.7 1024857.1 483857.1 292571.4
## [1443] 1844642.9 252142.9 430714.3 908428.6 624928.6 326300.0 356785.7
## [1450] 1243214.3 393142.9 352357.1 399285.7 519525.6 428157.1 460285.7
## [1457] 324214.3 442142.9 1154000.7 391000.0 485857.1 370785.7 390592.9
## [1464] 674000.0 570000.0 481785.7 793928.6 392135.7 1312857.1 366071.4
## [1471] 1395857.1 615000.0 2297857.1 545907.1 495857.1 1217928.6 225000.0
## [1478] 414500.0 498357.1 265285.7 348928.6 261214.3 366714.3 395107.1
## [1485] 894642.9 1209285.7 217285.7 451571.4 354285.7 314142.9 772857.1
## [1492] 240821.4 614000.0 368285.7 654714.3 447214.3 538142.9 315142.9
## [1499] 500928.4 222428.6 249642.9 383000.0 1164214.3 512142.7 1339785.7
## [1506] 305357.1 442278.6 549285.7 304714.3 522500.0 573564.3 282285.7
## [1513] 1178857.1 597357.1 520000.0 404285.6 365357.1 392571.4 380714.3
## [1520] 341357.1 188642.9 956428.6 695142.9 525857.1 413928.6 721500.0
## [1527] 456571.4 345642.9 416357.1 1125500.0 795285.7 240857.1 435285.7
## [1534] 1491428.6 612285.7 751357.1 448228.4 920142.9 602357.1 408357.1
## [1541] 268635.7 309428.6 561857.1 493642.9 435714.3 325857.1 430678.6
## [1548] 364157.1 450785.7 1005285.7 437928.6 434928.6 463214.3 202614.3
## [1555] 377142.9 567500.0 539000.0 504857.1 414000.0 2629214.3 1271942.9
## [1562] 473821.3 2935000.0 304285.7 463928.6 384071.4 370071.4 454285.7
## [1569] 376000.0 484748.4 315571.4 290571.4 501421.4 415000.0 380500.0
## [1576] 248000.0 876285.7 455071.4 375035.7 335785.7 686071.4 526142.9
## [1583] 1176862.7 605714.3 492714.3 341428.6 464500.0 729071.4 586714.3
## [1590] 3831785.7 252713.6 1743678.6 425192.1 438071.4 599857.1 593357.1
## [1597] 235142.9 396057.1 517785.7 481000.0 266028.6 633500.0 824442.9
## [1604] 1193214.3 490878.6 497435.7 397642.9 604245.0 579000.0 350850.0
## [1611] 627571.4 1089542.0 1196385.7 321250.0 476714.3 2290142.9 391214.3
## [1618] 1015571.4 276071.4 306571.4 1035714.3 467571.4 943463.6 418785.7
## [1625] 409642.9 536142.9 383272.9 663357.1 508571.4 384000.0 493792.9
## [1632] 431428.6 750857.1 494214.3 481714.3 441000.0 525000.0 503571.4
## [1639] 435207.1 344642.9 547142.9 548571.4 295635.7 895000.0 363293.3
## [1646] 417428.6 459928.6 418000.0 700428.6 290178.6 562428.6 477849.3
## [1653] 500000.0 574642.7 288812.5 323171.4 251028.6 468571.4 438928.6
## [1660] 412428.6 350428.6 400428.6 450285.7 560714.3 364857.1 516142.9
## [1667] 428500.0 252142.9 229071.4 1118000.0 291428.6 611428.6 396571.4
## [1674] 715857.1 540071.4 1472142.9 401785.7 602850.0 1197857.1 603447.7
## [1681] 334142.9 307500.0 547070.7 308000.0 725000.0 1369428.6 914785.7
## [1688] 614285.7 527142.9 610357.1 218928.6 359571.4 564571.4 246571.4
## [1695] 492714.3 280642.9 2236428.6 407571.4 650000.0 540785.7 1425000.0
## [1702] 632285.7 434571.4 4334999.9 440071.4 354214.3 328428.6 538214.3
## [1709] 689971.4 329785.7 479285.7 945357.1 411262.1 185392.9 215571.4
## [1716] 1239285.6 358250.0 492778.6 210471.4 381357.1 602142.9 1028857.1
## [1723] 388185.7 215857.1 603714.3 220642.9 386000.0 360135.7 2284285.6
## [1730] 566857.1 418528.6 638364.3 524357.1 557928.6 296428.6 536428.6
## [1737] 282785.7 383750.0 455142.9 427778.6 531142.9 491857.1 389714.3
## [1744] 482857.1 738000.0 394207.1 407042.9 832857.1 296642.9 1205000.0
## [1751] 667428.6 432142.9 364785.7 398142.9 259642.9 356857.1 564000.0
## [1758] 452285.7 432000.0 654959.9 267571.4 310285.7 516071.4 259142.9
## [1765] 436357.1 568356.4 400571.4 477085.7 298714.3 518057.1 378500.0
## [1772] 525571.4 233428.6 526892.9 1247857.1 499278.6 304142.9 367714.3
## [1779] 314922.9 628642.9 465785.7 586821.4 253357.1 403428.6 352000.0
## [1786] 1349285.7 574142.9 488571.4 1467142.9 398142.9 489571.4 351071.4
## [1793] 644571.4 813602.9 440071.4 1350285.7 396642.9 325000.0 445142.9
## [1800] 528285.7 347071.4 393285.7 892307.7 358142.9 448000.0 425571.4
## [1807] 526142.9 470000.0 513857.1 353285.7 272714.3 297500.0 521207.1
## [1814] 602857.1 300785.7 445428.6 345357.1 648042.9 359500.0 332142.9
## [1821] 234071.4 497571.4 343114.3 562857.1 481142.9 742857.1 429571.4
## [1828] 530571.4 425249.9 473214.3 395714.3 476357.1 723214.3 2305699.9
## [1835] 734479.1 524814.3 365357.1 559857.1 255357.1 796428.6 401857.1
## [1842] 407857.1 312657.1 363714.3 1050714.3 402500.0 473500.0 1539285.7
## [1849] 4110714.3 1451857.1 681571.4 899714.3 444300.0 931082.1 259285.7
## [1856] 750357.1 3473071.4 606500.0 446142.9 725571.4 337357.1 600714.3
## [1863] 368428.6 437714.3 599285.7 450107.1 537857.1 505000.0 382928.6
## [1870] 493064.3 848642.9 504857.1 815714.3 343000.0 295285.7 215714.3
## [1877] 1743571.4 654285.7 334142.9 675568.6 503142.9 946428.6 330857.1
## [1884] 643278.6 504142.9 893870.6 614642.9 384500.0 287392.9 283000.0
## [1891] 606428.6 554928.6 375857.1 405785.7 327214.3 346000.0 1362142.9
## [1898] 279125.3 288357.1 358214.3 439000.0 439478.6 697857.1 714071.4
## [1905] 346642.9 373214.3 329428.6 429428.6 615714.3 429285.7 302892.9
## [1912] 324142.9 319857.1 440785.7 345000.0 763571.4 472142.9 395928.6
## [1919] 410714.3 440642.9 812428.6 1092264.3 810278.6 1075357.1 585099.7
## [1926] 398571.4 287285.7 578578.6 313000.0 539357.1 388071.4 446357.1
## [1933] 432161.1 445357.1 335428.6 1216428.6 456571.4 409071.4 468071.4
## [1940] 377642.9 492857.1 370812.9 341428.6 215000.0 452500.0 444285.7
## [1947] 454714.3 583928.6 412992.9 647857.1 406778.4 415857.1 830714.3
## [1954] 641428.6 596257.1 447900.0 321142.9 708571.4 611928.6 515571.4
## [1961] 280992.9 2264271.4 390642.9 507285.7 1712857.1 468642.9 633500.0
## [1968] 428928.6 390142.9 274785.7 282857.1 308000.0 509142.9 450857.1
## [1975] 487214.3 222928.6 326500.0 832928.6 267214.3 475714.3 412214.3
## [1982] 318642.9 1234285.7 428477.9 697785.7 490285.7 239857.1 685964.3
## [1989] 658785.7 660250.0 608214.3 476802.9set.seed(1)
# 1 neuron
model4_brnn1 <- train(
price ~ total_floor_area+co2_emissions_current+average_income+latitude+longitude+num_tube_lines+num_rail_lines+population+energy_consumption_current+num_light_rail_lines+london_zone+crime_number+distance_to_station+property_type,
train_data,
method = "brnn",
trControl = control,
tuneGrid = expand.grid(neurons=1)
)## + Fold1: neurons=1
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9696 alpha= 2.4282 beta= 220.5269
## - Fold1: neurons=1
## + Fold2: neurons=1
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9637 alpha= 2.8012 beta= 208.7922
## - Fold2: neurons=1
## + Fold3: neurons=1
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9686 alpha= 2.4911 beta= 223.9209
## - Fold3: neurons=1
## + Fold4: neurons=1
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9717 alpha= 2.381 beta= 234.7429
## - Fold4: neurons=1
## + Fold5: neurons=1
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9711 alpha= 2.5081 beta= 234.9489
## - Fold5: neurons=1
## Aggregating results
## Fitting final model on full training set
## Number of parameters (weights and biases) to estimate: 18
## Nguyen-Widrow method
## Scaling factor= 0.7
## gamma= 17.9733 alpha= 2.4954 beta= 225.8695predictions <- predict(model4_brnn1,test_data)
brnn_results1<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
# 2 neurons
model4_brnn2 <- train(
price ~ total_floor_area+co2_emissions_current+average_income+latitude+longitude+num_tube_lines+num_rail_lines+population+energy_consumption_current+num_light_rail_lines+london_zone+crime_number+distance_to_station+property_type,
train_data,
method = "brnn",
trControl = control,
tuneGrid = expand.grid(neurons=2)
)## + Fold1: neurons=2
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000581
## gamma= 35.7822 alpha= 1.0744 beta= 299.4348
## - Fold1: neurons=2
## + Fold2: neurons=2
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000581
## gamma= 35.7793 alpha= 1.1629 beta= 291.9246
## - Fold2: neurons=2
## + Fold3: neurons=2
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000581
## gamma= 35.7633 alpha= 1.0083 beta= 326.387
## - Fold3: neurons=2
## + Fold4: neurons=2
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000581
## gamma= 35.8282 alpha= 0.9392 beta= 319.3186
## - Fold4: neurons=2
## + Fold5: neurons=2
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000581
## gamma= 35.7844 alpha= 1.0645 beta= 308.6356
## - Fold5: neurons=2
## Aggregating results
## Fitting final model on full training set
## Number of parameters (weights and biases) to estimate: 36
## Nguyen-Widrow method
## Scaling factor= 0.7000465
## gamma= 35.819 alpha= 1.0518 beta= 309.1937predictions <- predict(model4_brnn2,test_data)
brnn_results2<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
# 3 neurons
model4_brnn3 <- train(
price ~ total_floor_area+co2_emissions_current+average_income+latitude+longitude+num_tube_lines+num_rail_lines+population+energy_consumption_current+num_light_rail_lines+london_zone+crime_number+distance_to_station+property_type,
train_data,
method = "brnn",
trControl = control,
tuneGrid = expand.grid(neurons=3)
)## + Fold1: neurons=3
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000921
## gamma= 52.8702 alpha= 0.9782 beta= 322.566
## - Fold1: neurons=3
## + Fold2: neurons=3
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000921
## gamma= 53.2531 alpha= 1.0801 beta= 320.6918
## - Fold2: neurons=3
## + Fold3: neurons=3
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000921
## gamma= 52.913 alpha= 0.8304 beta= 335.2411
## - Fold3: neurons=3
## + Fold4: neurons=3
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000921
## gamma= 53.1251 alpha= 0.1129 beta= 385.9223
## - Fold4: neurons=3
## + Fold5: neurons=3
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000921
## gamma= 53.2037 alpha= 1.1323 beta= 320.8945
## - Fold5: neurons=3
## Aggregating results
## Fitting final model on full training set
## Number of parameters (weights and biases) to estimate: 54
## Nguyen-Widrow method
## Scaling factor= 0.7000737
## gamma= 53.0461 alpha= 1.0833 beta= 327.5292predictions <- predict(model4_brnn3,test_data)
brnn_results3<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
# 4 neurons
model4_brnn4 <- train(
price ~ total_floor_area+co2_emissions_current+average_income+latitude+longitude+num_tube_lines+num_rail_lines+population+energy_consumption_current+num_light_rail_lines+london_zone+crime_number+distance_to_station+property_type,
train_data,
method = "brnn",
trControl = control,
tuneGrid = expand.grid(neurons=4)
)## + Fold1: neurons=4
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7001162
## gamma= 69.8144 alpha= 1.0808 beta= 236.5593
## - Fold1: neurons=4
## + Fold2: neurons=4
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7001162
## gamma= 70.7664 alpha= 0.1171 beta= 378.9905
## - Fold2: neurons=4
## + Fold3: neurons=4
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7001162
## gamma= 70.6434 alpha= 0.2226 beta= 396.6563
## - Fold3: neurons=4
## + Fold4: neurons=4
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7001162
## gamma= 69.9449 alpha= 0.987 beta= 344.1757
## - Fold4: neurons=4
## + Fold5: neurons=4
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7001162
## gamma= 70.7362 alpha= 0.0987 beta= 430.2258
## - Fold5: neurons=4
## Aggregating results
## Fitting final model on full training set
## Number of parameters (weights and biases) to estimate: 72
## Nguyen-Widrow method
## Scaling factor= 0.7000929
## gamma= 70.971 alpha= 0.0944 beta= 390.3082predictions <- predict(model4_brnn4,test_data)
brnn_results4<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
# 5 neurons
model4_brnn5 <- train(
price ~ total_floor_area+co2_emissions_current+average_income+latitude+longitude+num_tube_lines+num_rail_lines+population+energy_consumption_current+num_light_rail_lines+london_zone+crime_number+distance_to_station+property_type,
train_data,
method = "brnn",
trControl = control,
tuneGrid = expand.grid(neurons=5)
)## + Fold1: neurons=5
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001349
## gamma= 86.4998 alpha= 0.1911 beta= 412.8044
## - Fold1: neurons=5
## + Fold2: neurons=5
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001349
## gamma= 87.6152 alpha= 0.6934 beta= 388.7802
## - Fold2: neurons=5
## + Fold3: neurons=5
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001349
## gamma= 87.6363 alpha= 0.2775 beta= 422.9289
## - Fold3: neurons=5
## + Fold4: neurons=5
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001349
## gamma= 87.1703 alpha= 0.3695 beta= 371.1612
## - Fold4: neurons=5
## + Fold5: neurons=5
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001349
## gamma= 85.9534 alpha= 0.0979 beta= 390.5327
## - Fold5: neurons=5
## Aggregating results
## Fitting final model on full training set
## Number of parameters (weights and biases) to estimate: 90
## Nguyen-Widrow method
## Scaling factor= 0.7001079
## gamma= 88.5332 alpha= 0.0754 beta= 418.5014predictions <- predict(model4_brnn5,test_data)
brnn_results5<-data.frame( RMSE = RMSE(predictions, test_data$price),
Rsquare = R2(predictions, test_data$price))
brnn_results<-data.frame(matrix(ncol = 3, nrow = 5))
x <- c("number_neuron", "RMSE", "Rsquare")
colnames(brnn_results) <- x
brnn_results$number_neuron[1] <- 1
brnn_results$number_neuron[2] <- 2
brnn_results$number_neuron[3] <- 3
brnn_results$number_neuron[4] <- 4
brnn_results$number_neuron[5] <- 5
brnn_results$RMSE[1] <- brnn_results1$RMSE
brnn_results$RMSE[2] <- brnn_results2$RMSE
brnn_results$RMSE[3] <- brnn_results3$RMSE
brnn_results$RMSE[4] <- brnn_results4$RMSE
brnn_results$RMSE[5] <- brnn_results5$RMSE
brnn_results$Rsquare[1] <- brnn_results1$Rsquare
brnn_results$Rsquare[2] <- brnn_results2$Rsquare
brnn_results$Rsquare[3] <- brnn_results3$Rsquare
brnn_results$Rsquare[4] <- brnn_results4$Rsquare
brnn_results$Rsquare[5] <- brnn_results5$Rsquarehighlight <- brnn_results %>%
filter(number_neuron==3) # optimal number of neurons
ggplot(data=brnn_results, aes(x=number_neuron, y=RMSE, group=1)) +
geom_line()+
geom_point()+
geom_point(data=highlight,
aes(x=number_neuron,y=RMSE),
color='red',
size=3) +
theme_minimal() +
labs(title = "Model achieved lowest RMSE at 3 neurons",
subtitle = "RMSE with different number of neurons",
x = "Number of neurons",
y = "RMSE")
ggplot(data=brnn_results, aes(x=number_neuron, y=Rsquare, group=1)) +
geom_line()+
geom_point()+
geom_point(data=highlight,
aes(x=number_neuron,y=Rsquare),
color='red',
size=3) +
theme_minimal() +
labs(title = "Model achieved highest Rsquared at 3 neurons",
subtitle = "Rsquared with different number of neurons",
x = "Number of neurons",
y = "Rsquared")
#you can also visualize the variable importance
importance <- varImp(model4_brnn3, scale=TRUE)
plot(importance)
#We can predict prices for out of sample data the same way
predictions_oos <- predict(model4_brnn3,london_house_prices_2019_out_of_sample)
predictions_oos## [1] 952270.3 317565.3 379754.7 445816.8 1005285.0 524024.1 819215.6
## [8] 321010.1 702625.7 507153.0 425614.6 1002260.1 303196.9 1202945.2
## [15] 352114.1 432980.8 321321.4 225412.4 754116.2 1089450.4 596919.2
## [22] 275968.5 217888.2 1397539.2 1167261.9 541072.2 265534.2 321049.0
## [29] 1210096.6 573645.7 500499.2 499085.4 540619.2 1014608.3 868909.8
## [36] 370710.5 690717.0 344391.3 882482.2 639360.8 336943.4 914781.5
## [43] 478668.8 822023.8 629943.3 401760.8 304906.5 595761.7 334035.5
## [50] 302484.6 418529.8 418272.9 443374.1 404364.8 609106.7 849584.9
## [57] 539830.2 461792.0 533922.9 294901.5 3404679.5 259395.0 314955.0
## [64] 779162.7 252355.7 223006.8 535077.0 776845.5 670941.4 304316.9
## [71] 711864.5 482876.3 359687.3 484432.9 295866.2 280870.2 307158.6
## [78] 356653.3 801945.8 318105.5 323874.8 461373.5 527936.9 650988.2
## [85] 1879058.3 2078459.2 284822.3 280582.8 782219.6 368687.5 408085.9
## [92] 882870.1 312981.1 1463842.3 497554.2 292481.7 222214.1 357756.9
## [99] 286068.1 543557.6 287795.7 310443.6 563525.9 440115.8 322964.5
## [106] 315389.5 247131.5 440105.2 476829.0 436171.3 552951.8 420598.7
## [113] 404758.9 336301.8 298231.0 1773569.0 399726.7 516154.9 268488.6
## [120] 482851.4 613666.2 328902.7 387097.5 771815.0 1007453.7 552234.2
## [127] 719626.9 478593.5 321304.9 393887.5 428806.6 373332.1 658647.7
## [134] 434887.7 284153.9 1500211.9 299904.4 410600.2 516523.7 849933.4
## [141] 417626.2 744587.6 264380.5 611003.3 443129.1 1173150.2 250814.4
## [148] 326295.1 269039.3 454928.3 514317.2 391664.5 315701.4 275344.2
## [155] 439330.5 254496.4 412188.9 1134279.0 376296.9 845754.8 336430.9
## [162] 287407.2 252921.4 365540.5 527286.7 1122193.5 1326604.5 612172.6
## [169] 553825.3 424635.8 373710.4 1766519.3 827421.3 175660.6 298401.2
## [176] 1614615.8 203400.2 713285.1 552255.7 375343.7 426021.4 382412.8
## [183] 1882128.8 435394.4 595035.8 603866.8 1631135.5 401555.0 277760.1
## [190] 380033.8 405324.3 407678.8 347130.1 329514.5 456391.2 286586.5
## [197] 235776.6 540270.9 571891.8 318567.8 459888.1 959824.3 505443.5
## [204] 252187.1 377668.6 237081.7 428835.3 698539.1 278744.3 361902.8
## [211] 322204.1 369320.7 755911.4 604214.7 770905.6 321553.0 356740.8
## [218] 494555.4 253274.5 486349.0 267249.2 329064.5 423765.4 574578.9
## [225] 212804.3 339401.6 636681.8 379836.5 256323.1 1020092.3 365911.9
## [232] 496186.5 415002.7 476334.8 1486484.1 1258586.7 432523.3 298184.9
## [239] 659289.4 274473.1 327739.4 442581.2 336721.7 1698300.5 896540.1
## [246] 374303.8 335539.0 567588.1 6259260.8 1164988.4 351454.8 1589882.7
## [253] 492241.9 388931.8 750886.6 428531.3 482304.7 2303966.4 308386.1
## [260] 919270.2 858468.9 547877.7 535846.3 472561.4 821058.3 664036.8
## [267] 269960.6 290729.5 1537340.0 582651.8 1073592.6 439631.0 234836.5
## [274] 449529.2 329638.2 1131590.7 405592.2 397394.9 1468056.4 384372.8
## [281] 514002.0 266638.6 601915.5 372185.7 845259.6 358013.0 904438.6
## [288] 342073.5 615986.0 1163534.6 287088.8 321675.6 1925739.0 799151.8
## [295] 705578.5 635224.5 309990.0 532646.3 350512.5 306725.7 4541320.2
## [302] 588014.9 295121.5 624430.7 1253147.3 340365.2 457459.1 324934.0
## [309] 332273.9 839949.4 681430.9 359189.5 445013.9 434230.8 925049.1
## [316] 931172.5 420156.2 1776186.2 468755.0 463111.5 699019.2 437186.2
## [323] 279314.0 547896.6 335555.3 1201315.1 262962.3 299768.3 536043.6
## [330] 896959.8 327718.7 374045.1 338819.9 708735.5 261770.1 493909.9
## [337] 328140.9 635489.3 286551.8 416422.0 351784.5 296342.5 483679.6
## [344] 1424817.7 668624.8 1114881.4 362212.3 657829.3 296099.1 285477.3
## [351] 474220.3 777359.3 348433.5 370987.5 388972.4 438373.7 398327.4
## [358] 377871.5 296882.1 467638.2 503993.8 432993.7 246062.8 371228.9
## [365] 547966.7 426732.8 566951.7 556247.8 451658.1 437250.0 841913.6
## [372] 987163.2 629377.5 239904.7 697850.4 348564.6 586092.7 516455.2
## [379] 414565.1 363875.7 411827.2 285635.2 287483.1 1489122.3 349558.3
## [386] 380234.3 456724.9 340719.2 769045.8 548569.7 297558.1 483914.8
## [393] 378903.0 848653.9 301322.9 341391.1 473291.8 681865.5 314141.9
## [400] 333954.4 808312.2 1043901.3 573604.7 1038324.6 1009707.9 394448.0
## [407] 268657.4 319986.8 502121.0 569077.0 584894.6 445618.3 713915.8
## [414] 363945.5 294841.4 264914.6 362792.3 584304.2 474155.3 458702.3
## [421] 869963.7 446189.9 324190.7 369913.6 516431.8 496971.3 466425.6
## [428] 236690.6 532464.6 460785.1 438534.0 1192984.3 260240.8 1589838.9
## [435] 523289.2 303205.0 575667.2 782758.9 454708.9 212174.6 746406.7
## [442] 397676.7 856020.7 348089.3 346488.0 484265.1 973991.6 613861.4
## [449] 646400.0 636517.0 655189.7 507249.4 475364.6 1512761.3 473063.5
## [456] 230025.5 1386644.5 389254.1 341449.8 1345053.4 1581408.9 199599.8
## [463] 328429.3 248505.2 1004470.1 418029.9 432756.9 434012.7 707333.8
## [470] 632856.3 337698.3 363931.5 3043094.2 493004.0 282226.0 286372.9
## [477] 627187.6 437467.0 413185.0 666950.4 355970.5 899881.7 533912.4
## [484] 381265.0 375500.5 440838.9 901731.3 271277.6 438792.5 1807281.7
## [491] 2066457.2 316642.8 247111.5 924533.7 353504.0 688878.3 414152.3
## [498] 336217.5 901226.5 773916.1 541557.3 336519.9 801773.8 374190.5
## [505] 346529.5 318708.1 473358.8 546329.2 294670.0 496501.4 347561.2
## [512] 612589.0 1514504.3 473238.2 1150806.2 387084.7 346547.3 267162.4
## [519] 518500.9 590295.4 201956.1 412280.2 422842.0 350095.3 630086.5
## [526] 286300.0 3673393.2 280777.2 651366.3 417993.2 460470.3 345887.6
## [533] 402573.0 413872.3 554749.7 765805.5 379541.6 1156133.2 410077.0
## [540] 283391.2 426940.0 597906.3 1394323.0 483483.9 613997.2 385832.0
## [547] 771747.6 331459.9 644739.0 1110058.3 375606.6 2029663.3 403060.0
## [554] 1045294.3 393163.4 1609386.6 310941.1 544346.4 405657.3 290470.8
## [561] 376709.0 254065.4 1231416.3 681120.4 265605.5 1209686.8 815107.3
## [568] 397935.0 250443.6 486436.8 445519.1 456750.4 472240.1 1636060.1
## [575] 332762.0 274703.7 282076.9 810607.0 408469.9 392044.4 325366.3
## [582] 357989.0 672927.0 271490.3 1474661.7 258247.9 522192.7 331448.2
## [589] 220036.9 1065544.0 405281.4 307252.7 988675.9 271030.2 648205.5
## [596] 1218000.2 490828.9 456579.2 476127.7 321502.7 333780.8 507477.7
## [603] 448575.4 351284.1 838734.8 1638641.4 531138.3 613630.3 438939.6
## [610] 576392.6 399281.0 327778.4 447251.3 314506.4 1216023.3 310628.6
## [617] 848248.0 474594.1 340411.2 487102.8 669507.0 401612.9 423527.3
## [624] 440734.5 553471.7 174796.2 248214.9 414763.3 392334.4 674684.2
## [631] 445182.3 276059.3 477577.6 1568154.5 618387.9 335483.2 245573.8
## [638] 268508.4 502941.7 647756.6 322288.0 212235.1 302590.4 495106.6
## [645] 582719.1 309742.7 502061.5 417403.2 1425766.6 782169.5 654829.5
## [652] 360736.1 306846.1 489520.2 403393.2 255943.2 1012550.1 673438.1
## [659] 365119.7 458826.6 326538.6 2052251.7 298214.6 324945.8 627184.3
## [666] 365957.5 641387.7 2598578.5 519027.8 382016.6 970608.7 689966.3
## [673] 1547024.7 515899.6 395935.9 318920.2 1608154.4 491639.4 252871.5
## [680] 682101.6 506217.6 533307.0 428893.3 479111.7 484466.9 318354.9
## [687] 478944.8 421292.3 434609.1 477054.9 428658.1 403225.3 553994.3
## [694] 336249.4 333878.3 239472.5 836008.8 291437.3 2208719.6 2243427.2
## [701] 426985.0 362127.0 380261.8 359202.9 457065.9 402097.0 321777.0
## [708] 309061.5 555598.2 474453.8 246832.2 418845.6 525763.0 385556.6
## [715] 388904.0 311716.4 871716.4 450868.1 402205.8 365016.1 353406.2
## [722] 501249.2 396860.0 421763.1 533578.0 531371.8 268980.6 450909.8
## [729] 352111.9 645390.1 376896.4 436009.0 538094.6 265729.8 593537.0
## [736] 1560632.4 468788.9 540668.9 511699.9 366409.6 300997.9 606945.8
## [743] 331830.9 334461.3 280674.0 416441.5 208008.0 308539.2 1331192.9
## [750] 850389.4 1956909.8 459915.9 421894.1 472531.9 699349.8 422684.9
## [757] 330225.7 718620.9 315987.1 393785.8 858647.9 410333.2 332174.3
## [764] 426391.5 432278.3 911992.9 682243.7 718369.8 705839.5 581965.3
## [771] 529582.2 651347.1 339115.2 775636.9 459523.0 384495.2 1821100.2
## [778] 385618.8 355274.0 385711.9 452453.1 420278.2 449787.6 330573.9
## [785] 2232944.1 606120.8 447538.5 479550.5 530350.8 376724.5 327018.5
## [792] 765900.9 765866.9 900524.1 330110.6 445095.0 251027.3 853456.6
## [799] 707668.2 512793.2 956456.4 1169069.6 383773.4 333792.8 588542.0
## [806] 453463.6 1231711.6 428568.7 1918351.4 500081.0 626513.4 276095.4
## [813] 480545.8 378341.7 282134.9 511811.9 694731.5 707627.0 469804.2
## [820] 282607.2 1003166.6 3009493.2 198632.1 1251363.9 584873.5 467233.3
## [827] 353099.5 499686.7 621131.3 460255.2 333609.0 524000.7 438917.3
## [834] 336423.5 411212.4 665537.3 126381.8 489417.7 364574.6 559669.4
## [841] 446461.2 331975.0 926793.4 348933.6 949910.7 513241.5 649436.8
## [848] 401796.1 941032.0 1164894.4 1275868.4 614736.2 401232.9 488229.1
## [855] 419960.2 429905.1 459356.4 273772.3 442251.3 305940.3 516137.9
## [862] 508948.5 1885139.7 546635.0 496356.1 425944.6 1219039.2 1612067.5
## [869] 251199.7 545879.8 636981.4 417111.9 290277.6 546174.8 375407.4
## [876] 319630.9 450827.3 441065.2 1452800.2 647678.8 337814.1 1101529.8
## [883] 298855.9 274546.9 288531.4 356454.2 388012.0 603110.0 356197.4
## [890] 313782.7 668157.4 375597.9 568913.4 839627.7 775957.9 357640.9
## [897] 462474.6 402862.6 614092.9 556189.4 471894.0 305843.2 1027491.0
## [904] 345112.8 529415.4 465701.1 588986.5 245145.5 477828.3 240773.1
## [911] 453475.1 301121.6 401996.7 293149.1 584159.6 289968.3 1079802.2
## [918] 333312.1 652218.9 350782.8 366454.6 674099.3 283761.0 331746.0
## [925] 339644.5 483826.8 475315.2 683521.8 383217.7 565569.6 254445.6
## [932] 340908.7 696165.8 330918.2 268748.7 303429.0 371634.6 1213046.9
## [939] 257043.1 754661.8 670674.1 488130.7 325366.9 346456.7 523738.4
## [946] 309705.0 331372.8 518658.3 639557.2 486120.2 227701.7 334275.1
## [953] 833063.0 471935.2 358034.2 328844.3 539899.3 452637.8 507020.1
## [960] 252126.4 418981.0 389259.4 1282910.0 338800.9 888590.3 558279.8
## [967] 378854.8 1418209.5 1656636.9 1277964.3 960151.9 815427.6 337877.0
## [974] 348939.7 420449.1 417896.5 518137.5 745491.5 358922.3 980879.2
## [981] 476863.4 413065.7 318756.9 658983.9 2155805.9 453837.1 483130.5
## [988] 792466.2 364190.6 266274.7 960445.2 439749.5 562244.1 1738560.9
## [995] 482105.6 403952.8 563940.7 253967.3 491346.8 520543.3 399642.3
## [1002] 638366.7 609420.1 474274.2 386948.6 331913.3 324158.1 275220.7
## [1009] 301572.6 789583.5 451754.2 352620.5 987586.4 350443.9 315203.4
## [1016] 422984.0 313663.1 346518.9 550843.2 314439.7 307694.4 428993.0
## [1023] 429562.9 380732.8 457358.4 591618.6 1742088.6 347381.3 344429.9
## [1030] 797528.9 425430.9 1269854.3 343507.8 262333.7 412714.4 462770.0
## [1037] 488354.1 252878.9 501278.3 370426.8 246416.5 634386.4 364692.1
## [1044] 317535.5 385210.2 543169.6 223118.9 267423.7 343577.6 350037.1
## [1051] 856068.6 415417.6 367095.8 585734.3 539849.9 1972940.4 415507.2
## [1058] 385592.4 2290239.8 381390.3 1414377.3 902088.8 400674.0 475806.0
## [1065] 591841.3 651369.1 410753.3 1348124.7 310866.6 376274.6 2140253.9
## [1072] 343466.1 284314.6 429774.9 348512.3 1022550.1 366277.7 597485.7
## [1079] 252688.5 400098.1 362371.5 485431.6 418662.0 765037.4 421862.7
## [1086] 584399.0 361722.2 796024.3 873556.4 256429.5 739062.4 438028.2
## [1093] 208671.3 224710.0 920635.5 535586.0 392126.9 338062.1 349251.8
## [1100] 229684.4 393766.7 1417137.6 725780.4 766613.1 363781.6 584717.1
## [1107] 533205.1 1638773.0 343788.4 674060.9 388837.6 2286205.7 725589.4
## [1114] 449604.7 362497.3 467763.3 612284.3 565969.1 379967.4 349410.4
## [1121] 316379.7 419679.7 244714.6 888450.3 624014.5 390210.6 384695.5
## [1128] 609450.6 504789.0 344826.6 538362.2 373010.1 251809.6 310377.1
## [1135] 1807394.1 653394.9 601264.9 651834.1 614163.9 304059.4 368954.3
## [1142] 390287.5 710153.1 225782.1 350230.2 1153728.1 464516.9 814204.3
## [1149] 276205.6 258343.6 691570.0 1193253.4 285070.9 380072.0 294720.7
## [1156] 2983921.3 409117.2 549017.7 396005.1 312883.5 776572.3 861717.6
## [1163] 480916.1 855360.2 265854.2 513616.3 434777.3 507344.2 493605.9
## [1170] 450378.2 778537.8 316976.6 444944.4 515472.5 695519.3 479083.8
## [1177] 269556.8 866590.0 546835.5 847071.3 635126.3 409149.3 444324.2
## [1184] 883582.8 475001.7 294626.5 1589299.7 316754.2 325824.2 244021.5
## [1191] 546077.1 242849.2 313261.1 1326415.2 386656.6 1584409.7 599805.4
## [1198] 447242.4 527432.9 468920.0 644652.5 1372339.3 398561.7 356688.5
## [1205] 553506.1 277804.1 803661.6 430125.3 736849.1 407619.5 274143.9
## [1212] 828668.9 291107.2 771673.5 255119.8 409877.1 577403.2 574074.4
## [1219] 453444.9 461272.7 497472.0 924989.8 310598.2 518586.9 395395.0
## [1226] 272736.3 904939.9 950268.6 438819.0 739853.2 409578.9 347960.6
## [1233] 319879.7 247195.9 333380.5 1036242.4 410849.9 542859.6 499313.8
## [1240] 313458.5 1531261.0 772719.6 442704.7 501905.4 956709.1 423021.3
## [1247] 331249.6 510037.3 947286.7 329838.3 726905.2 349838.4 1028606.3
## [1254] 391398.6 317647.4 893584.9 371710.2 943450.1 454107.1 563652.2
## [1261] 248389.5 458439.3 507684.5 390723.7 340673.9 468469.4 432703.2
## [1268] 517176.3 850976.4 291990.2 705638.7 341794.1 386415.8 535639.2
## [1275] 419827.6 3885929.1 344554.6 657479.1 451103.8 719542.8 337218.5
## [1282] 312944.2 270432.5 618871.6 433703.5 1375393.8 487576.7 637579.8
## [1289] 1532778.2 330462.8 383010.5 587517.8 440923.9 867587.0 704967.6
## [1296] 450973.3 533072.4 417385.7 557238.2 1250509.0 392949.3 343555.5
## [1303] 400488.9 275447.2 659892.8 609283.2 512054.3 262351.8 1111214.0
## [1310] 5243791.1 341138.8 314530.9 619872.1 1413268.0 430430.1 407702.4
## [1317] 1200441.4 622914.3 394023.5 680673.3 384204.9 407132.5 291137.1
## [1324] 760451.5 717876.8 737046.8 766602.6 431602.3 942575.7 509202.3
## [1331] 318796.8 439604.4 349043.1 880337.2 308598.9 307446.6 334741.5
## [1338] 351080.1 1445983.6 639717.3 455135.5 2418336.2 315215.3 750322.5
## [1345] 719025.5 762520.2 636300.7 440967.0 425797.2 391124.5 572786.9
## [1352] 310242.9 661431.7 337722.8 404931.3 974489.1 435306.3 475545.5
## [1359] 2168933.5 519926.2 983203.8 576677.5 383142.1 292811.4 364885.0
## [1366] 310672.8 385746.8 266515.4 389433.3 991054.0 416351.0 1014175.9
## [1373] 263882.5 293206.3 690617.4 527489.2 441863.0 338452.9 423668.7
## [1380] 455920.9 291437.3 1188449.3 273984.7 314733.3 332124.2 426952.4
## [1387] 1043598.3 424350.3 425753.9 1156511.9 1888458.9 489281.2 230512.3
## [1394] 402188.7 563165.1 360435.8 345285.9 447660.9 503711.6 643158.7
## [1401] 327521.5 359552.7 411068.3 494297.9 391935.1 811897.7 214702.0
## [1408] 649286.0 426801.8 901923.7 348106.6 263328.1 297405.4 340690.0
## [1415] 1273380.8 417760.4 345354.1 849488.8 2135605.7 322120.7 432310.6
## [1422] 362459.4 343303.4 575217.5 545344.1 373245.1 599164.0 283348.6
## [1429] 296141.2 414095.1 1411578.3 348191.7 755563.9 365881.9 326204.4
## [1436] 360813.2 314882.3 585812.5 394946.3 952453.1 439390.0 295394.3
## [1443] 2538361.8 262032.6 416627.2 1342184.8 883679.9 231392.9 350103.4
## [1450] 1483429.5 310738.6 310448.8 357637.4 487788.9 434721.1 399464.2
## [1457] 497463.4 467935.0 1181841.3 397324.0 527638.5 369597.1 393819.7
## [1464] 474295.4 547244.9 398588.5 968438.1 401243.9 1581643.8 380458.9
## [1471] 1526165.6 657338.6 2965385.9 464678.1 442027.6 1107659.7 282785.7
## [1478] 378775.7 424949.8 325198.5 323276.8 290510.8 341947.2 643156.1
## [1485] 821815.2 1077921.1 289190.1 493986.8 322104.9 320368.0 776820.6
## [1492] 337798.6 568916.5 371636.2 506103.4 371373.7 422799.8 358488.5
## [1499] 454407.3 306713.5 220871.9 390151.3 1249446.0 368233.6 885794.6
## [1506] 298760.4 401613.6 457784.7 313441.8 892727.3 446224.0 261368.1
## [1513] 1104416.8 580781.5 601668.6 310088.6 366050.7 442404.7 333778.8
## [1520] 381475.7 216934.0 813363.9 914506.5 495096.9 334593.9 835385.9
## [1527] 464213.2 320743.4 481554.1 1224411.4 934875.9 269843.0 390959.0
## [1534] 1965905.1 533193.0 951547.0 336067.7 865748.5 453946.8 386736.8
## [1541] 271078.2 327246.8 463395.2 551138.6 398926.7 220400.8 438386.9
## [1548] 275339.1 430776.3 1248463.0 464354.2 360740.1 429114.8 274231.6
## [1555] 357041.0 369400.8 574124.0 582962.6 417060.4 2178388.6 1438276.1
## [1562] 817333.7 2320163.2 320080.1 485645.1 368954.5 332001.1 391619.4
## [1569] 358175.0 359697.6 326433.8 307913.3 644114.4 373076.1 347500.9
## [1576] 273995.9 1120703.7 394855.4 433480.6 358225.0 579182.9 621083.7
## [1583] 1318793.7 710316.2 448933.9 336620.7 414667.5 747732.8 461901.9
## [1590] 9066238.8 293694.3 1440169.6 375220.7 551054.0 690642.6 866746.4
## [1597] 261608.7 367174.9 487273.5 407931.1 290751.4 722796.4 865569.9
## [1604] 1093641.1 412845.2 581617.1 348281.3 611421.9 665518.9 441947.4
## [1611] 772232.2 1125361.1 1246799.8 400780.2 480455.5 1791418.7 422843.3
## [1618] 1137142.0 329595.0 337422.2 954841.0 536036.9 1251008.2 628398.4
## [1625] 390500.6 701848.6 369948.9 648646.1 511642.7 344129.1 475412.8
## [1632] 439324.5 651720.9 552739.9 599600.8 561173.2 407261.0 555295.7
## [1639] 449355.3 273611.9 497760.1 949101.8 332450.3 802496.7 404212.1
## [1646] 343362.4 401303.9 636425.2 659908.2 279015.5 467760.4 570255.3
## [1653] 411256.7 589606.8 245216.6 271640.3 241306.5 475409.1 408086.8
## [1660] 458919.6 382418.9 414531.1 382773.1 726700.1 356130.7 470373.7
## [1667] 328216.8 293138.7 252928.9 766213.4 250370.9 525835.7 382331.1
## [1674] 698206.8 477206.2 1346537.5 408794.5 654620.4 1233793.1 672836.7
## [1681] 352711.7 285176.4 509361.4 326801.3 986679.2 1305134.8 883445.5
## [1688] 501457.2 476648.2 400560.1 267616.4 419169.6 593810.5 188880.7
## [1695] 729599.2 279218.0 3078313.2 354667.3 454753.5 527905.1 1372329.8
## [1702] 505391.2 410444.7 3178461.6 397416.6 372008.7 318669.5 543758.0
## [1709] 625722.3 350055.8 352687.1 1019398.6 386092.6 286371.5 219618.4
## [1716] 1369332.0 328979.3 418578.6 287188.6 467827.5 645356.4 1322924.8
## [1723] 540960.3 278871.6 610118.8 183215.1 439734.2 367211.8 1809007.6
## [1730] 539354.9 498329.3 620419.3 403914.9 473315.5 324585.5 345409.1
## [1737] 372658.4 400641.0 473121.3 432112.0 452665.9 511056.0 383316.3
## [1744] 368628.9 652549.2 395324.9 307629.7 842886.7 281686.2 1523207.4
## [1751] 603233.6 365427.7 409753.6 379994.5 266655.7 271135.9 366159.1
## [1758] 452123.6 601885.3 922193.2 334705.4 323442.4 434010.9 264855.7
## [1765] 514822.0 577231.0 426759.4 583402.1 318104.7 517810.1 322612.9
## [1772] 378288.1 250034.5 471398.4 1032354.9 413048.7 288009.7 435316.3
## [1779] 295036.4 800710.9 378467.5 625909.0 309368.9 421766.3 355328.3
## [1786] 1161779.2 618780.2 419860.8 1719764.3 346559.5 413960.9 371677.2
## [1793] 693699.7 849751.6 419557.1 1413306.6 399331.4 331660.9 430230.6
## [1800] 624649.7 398358.4 391471.7 652253.4 382343.0 412139.3 509026.1
## [1807] 838651.1 585708.1 449487.0 389660.1 354640.7 294898.2 450237.9
## [1814] 697382.1 337645.9 413492.9 495213.0 906761.2 290507.2 323952.9
## [1821] 237913.6 388528.2 407169.6 695815.2 516548.5 613010.8 415192.0
## [1828] 435661.3 469475.8 402556.1 485902.4 416766.6 493745.1 2102127.7
## [1835] 691917.0 487997.7 341671.3 405562.2 291435.1 786567.8 376077.6
## [1842] 435274.1 348199.7 349227.9 1094768.2 461664.0 389550.8 1409702.9
## [1849] 3888086.9 1396077.4 769454.2 924684.5 349520.1 1141665.4 224862.6
## [1856] 1181777.4 2296568.3 479770.6 411485.8 707882.8 395227.8 431183.7
## [1863] 482515.4 356266.4 889083.8 607356.8 480666.1 393077.3 234670.5
## [1870] 406005.4 1026533.4 534039.2 708890.9 328608.3 222198.8 266482.9
## [1877] 1261887.4 388662.9 325743.9 793681.9 465965.4 736151.3 295733.4
## [1884] 732813.8 490047.9 1060574.4 750470.7 398753.4 289297.0 277476.6
## [1891] 726362.9 784699.0 342888.7 348703.7 316820.3 319213.1 959820.9
## [1898] 326072.9 281497.2 322870.9 386550.3 465574.3 567362.2 747396.2
## [1905] 284123.4 377522.9 369854.6 478735.9 581848.8 473446.5 427099.9
## [1912] 318492.7 344618.9 391917.6 280426.4 955021.8 421658.3 392796.4
## [1919] 417164.2 469075.1 676711.1 1135803.6 939656.0 1126987.4 458920.4
## [1926] 393038.8 248004.9 763558.1 295410.6 669937.3 378507.6 422885.4
## [1933] 470527.8 472236.7 460692.2 966261.7 620883.0 368139.3 447594.0
## [1940] 322642.3 431292.4 346518.3 321545.0 248108.7 437457.7 676038.2
## [1947] 421958.0 589680.5 595917.3 567820.5 420656.8 414966.6 607231.7
## [1954] 674084.8 815537.5 368632.7 360034.3 761989.3 581457.9 418074.2
## [1961] 272900.7 1835035.5 340352.7 500084.6 1346547.8 577365.3 515369.5
## [1968] 409667.0 414047.5 286524.9 278784.1 309390.2 501577.6 365015.2
## [1975] 362179.8 253538.9 337784.7 780226.3 318566.9 427843.3 429396.0
## [1982] 316472.2 1512501.0 386087.1 594203.3 445518.0 291725.0 503167.1
## [1989] 921803.2 967154.3 811225.4 392957.67 Stacking
Use stacking to ensemble your algorithms.
set.seed(1) # to return the same result in the following chunks
train_predict_lm <- predict(model1_lm,train_data)
train_predict_tree <- predict(model2_tree,train_data)
train_predict_knn <- predict(model3_knn,train_data)
train_predict_brnn <- predict(model4_brnn3,train_data)
stacking_train <- as.data.frame(cbind(train_predict_lm,train_predict_tree,train_predict_knn,train_predict_brnn, train_data$price))
names(stacking_train)[5] <- "price"
#we are going to train the combiner model and report the results using k-fold cross validation
combiner <- train(
price ~ .,
stacking_train,
method = "lm",
trControl = control,
metric = "RMSE")## + Fold1: intercept=TRUE
## - Fold1: intercept=TRUE
## + Fold2: intercept=TRUE
## - Fold2: intercept=TRUE
## + Fold3: intercept=TRUE
## - Fold3: intercept=TRUE
## + Fold4: intercept=TRUE
## - Fold4: intercept=TRUE
## + Fold5: intercept=TRUE
## - Fold5: intercept=TRUE
## Aggregating results
## Fitting final model on full training setstacking_train$train_predictions <- predict(combiner,stacking_train)
stacking_train_result<-data.frame( RMSE = RMSE(stacking_train$train_predictions, stacking_train$price),
Rsquare = R2(stacking_train$train_predictions, stacking_train$price))
stacking_train_result # performance of combiner model on training data## RMSE Rsquare
## 1 194822.8 0.8623335test_predict_lm <- predict(model1_lm,test_data)
test_predict_tree <- predict(model2_tree,test_data)
test_predict_knn <- predict(model3_knn,test_data)
test_predict_brnn <- predict(model4_brnn3,test_data)
stacking_test <- as.data.frame(cbind(test_predict_lm,test_predict_tree,test_predict_knn,test_predict_brnn))
colnames(stacking_test) = colnames(stacking_train)[1:4]
stacking_test$test_predictions <- predict(combiner,stacking_test)
stacking_test_result<-data.frame( RMSE = RMSE(stacking_test$test_predictions, test_data$price),
Rsquare = R2(stacking_test$test_predictions, test_data$price))
stacking_test_result # performance of combiner model on testing data## RMSE Rsquare
## 1 217744.1 0.81344218 Pick investments
In this section you should use the best algorithm you identified to choose 200 properties from the out of sample data.
set.seed(1) # to return the same result in the following chunks
numchoose=200 # number of houses to invest in
oos<-london_house_prices_2019_out_of_sample
oos_predict_lm <- predict(model1_lm,oos)
oos_predict_tree <- predict(model2_tree,oos)
oos_predict_knn <- predict(model3_knn,oos)
oos_predict_brnn <- predict(model4_brnn3,oos)
stacking_oos <- as.data.frame(cbind(oos_predict_lm,oos_predict_tree,oos_predict_knn,oos_predict_brnn))
colnames(stacking_oos) = colnames(stacking_train)[1:4]
oos$predict <- predict(combiner,stacking_oos)
#predict the value of houses
selection <- oos %>%
mutate(profit = (predict - asking_price)/asking_price) %>%
slice_max(profit, n = numchoose) %>% # Choose the 200 houses to invest
select(ID)
selection$buy <- 1
oos<-read.csv("test_data_assignment.csv")
oos <- oos %>%
left_join(selection, by = "ID")
oos$buy[is.na(oos$buy)] <- 0
#output your choices. Change the name of the file to your "lastname_firstname.csv"
write.csv(oos,"Ding_Linli.csv")